Problem Set #2

In [1]:
# Importing relevant libraries
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

1.1.2. Now repeat the previous exercise on computer. Choose at least one stopping criterion and it's parameter e, and the bail-out number of iterations, R. Implement the above as an algorithm that at each step prints out the x value, the function's value f(x) and gradient rf(x). At the end it should print the solution, and also how many iterations it took for the loop to converge (to finish).

In [2]:
# Function to calculate the value of f(x)
def f(x):
    return (-x*x)
In [3]:
# Function to calculate the value of gradient
def grad(x):
    return (-2*x)
In [4]:
# Setting x0
x0 = 1

# Learning rate
R = 0.1

# Calculating x1
x1 = x0 + R*grad(x0)
In [5]:
print("Location of maximum for fx:",0)
print("Value of f(x0):",f(x0))
print("Value of gradient at x0:",grad(x0))
print ("Value of x1",x1)
print ("Yes, we moved closer to the maximum with x moving from 1 to 0.8")
Location of maximum for fx: 0
Value of f(x0): -1
Value of gradient at x0: -2
Value of x1 0.8
Yes, we moved closer to the maximum with x moving from 1 to 0.8
In [6]:
# Calculating number of iterations and value of the solution
def converge(x0,R):
    l=[]
    i = 0

    while True:
        a = x0
        l.append(x0 + (R)*grad(x0))
        x0 = l[i]
    
        if abs(grad(f(x0))) < (0.000001) or i > 50:
            print ("\n\nTotal number of iterations:",len(l)-1)
            print ("Solution value:",f(a))
            break
        else:
            fx = f(x0)
            gradient = grad(x0)
            print("Value of function f(x0):",fx)
            print("Value of the gradient at x0:",gradient)
            print("Value of x0:",a)
            print("Value of x1:",l[i])
            i = i + 1
In [7]:
converge(0.1,0.1)
Value of function f(x0): -0.0064
Value of the gradient at x0: -0.16
Value of x0: 0.1
Value of x1: 0.08
Value of function f(x0): -0.004096
Value of the gradient at x0: -0.128
Value of x0: 0.08
Value of x1: 0.064
Value of function f(x0): -0.00262144
Value of the gradient at x0: -0.1024
Value of x0: 0.064
Value of x1: 0.0512
Value of function f(x0): -0.0016777216000000003
Value of the gradient at x0: -0.08192
Value of x0: 0.0512
Value of x1: 0.04096
Value of function f(x0): -0.0010737418240000004
Value of the gradient at x0: -0.06553600000000001
Value of x0: 0.04096
Value of x1: 0.032768000000000005
Value of function f(x0): -0.0006871947673600003
Value of the gradient at x0: -0.05242880000000001
Value of x0: 0.032768000000000005
Value of x1: 0.026214400000000006
Value of function f(x0): -0.00043980465111040015
Value of the gradient at x0: -0.04194304000000001
Value of x0: 0.026214400000000006
Value of x1: 0.020971520000000004
Value of function f(x0): -0.00028147497671065613
Value of the gradient at x0: -0.03355443200000001
Value of x0: 0.020971520000000004
Value of x1: 0.016777216000000005
Value of function f(x0): -0.00018014398509481994
Value of the gradient at x0: -0.026843545600000008
Value of x0: 0.016777216000000005
Value of x1: 0.013421772800000004
Value of function f(x0): -0.00011529215046068476
Value of the gradient at x0: -0.021474836480000006
Value of x0: 0.013421772800000004
Value of x1: 0.010737418240000003
Value of function f(x0): -7.378697629483823e-05
Value of the gradient at x0: -0.017179869184000003
Value of x0: 0.010737418240000003
Value of x1: 0.008589934592000002
Value of function f(x0): -4.722366482869647e-05
Value of the gradient at x0: -0.013743895347200002
Value of x0: 0.008589934592000002
Value of x1: 0.006871947673600001
Value of function f(x0): -3.022314549036574e-05
Value of the gradient at x0: -0.010995116277760002
Value of x0: 0.006871947673600001
Value of x1: 0.005497558138880001
Value of function f(x0): -1.934281311383407e-05
Value of the gradient at x0: -0.008796093022208
Value of x0: 0.005497558138880001
Value of x1: 0.004398046511104
Value of function f(x0): -1.2379400392853805e-05
Value of the gradient at x0: -0.007036874417766401
Value of x0: 0.004398046511104
Value of x1: 0.0035184372088832004
Value of function f(x0): -7.922816251426435e-06
Value of the gradient at x0: -0.00562949953421312
Value of x0: 0.0035184372088832004
Value of x1: 0.00281474976710656
Value of function f(x0): -5.070602400912918e-06
Value of the gradient at x0: -0.004503599627370496
Value of x0: 0.00281474976710656
Value of x1: 0.002251799813685248
Value of function f(x0): -3.245185536584268e-06
Value of the gradient at x0: -0.003602879701896397
Value of x0: 0.002251799813685248
Value of x1: 0.0018014398509481986
Value of function f(x0): -2.0769187434139313e-06
Value of the gradient at x0: -0.0028823037615171177
Value of x0: 0.0018014398509481986
Value of x1: 0.0014411518807585589
Value of function f(x0): -1.3292279957849161e-06
Value of the gradient at x0: -0.002305843009213694
Value of x0: 0.0014411518807585589
Value of x1: 0.001152921504606847
Value of function f(x0): -8.507059173023462e-07
Value of the gradient at x0: -0.0018446744073709553
Value of x0: 0.001152921504606847
Value of x1: 0.0009223372036854776
Value of function f(x0): -5.444517870735017e-07
Value of the gradient at x0: -0.0014757395258967643
Value of x0: 0.0009223372036854776
Value of x1: 0.0007378697629483821


Total number of iterations: 22
Solution value: -5.444517870735017e-07

1.1.3. Experiment with different starting values, learning rates and stopping parameter values. Comment and explain your findings.

In [8]:
converge(1,0.1)
Value of function f(x0): -0.6400000000000001
Value of the gradient at x0: -1.6
Value of x0: 1
Value of x1: 0.8
Value of function f(x0): -0.4096
Value of the gradient at x0: -1.28
Value of x0: 0.8
Value of x1: 0.64
Value of function f(x0): -0.262144
Value of the gradient at x0: -1.024
Value of x0: 0.64
Value of x1: 0.512
Value of function f(x0): -0.16777216
Value of the gradient at x0: -0.8192
Value of x0: 0.512
Value of x1: 0.4096
Value of function f(x0): -0.10737418240000002
Value of the gradient at x0: -0.65536
Value of x0: 0.4096
Value of x1: 0.32768
Value of function f(x0): -0.06871947673600003
Value of the gradient at x0: -0.5242880000000001
Value of x0: 0.32768
Value of x1: 0.26214400000000004
Value of function f(x0): -0.04398046511104002
Value of the gradient at x0: -0.4194304000000001
Value of x0: 0.26214400000000004
Value of x1: 0.20971520000000005
Value of function f(x0): -0.02814749767106561
Value of the gradient at x0: -0.33554432000000006
Value of x0: 0.20971520000000005
Value of x1: 0.16777216000000003
Value of function f(x0): -0.018014398509481992
Value of the gradient at x0: -0.26843545600000007
Value of x0: 0.16777216000000003
Value of x1: 0.13421772800000004
Value of function f(x0): -0.011529215046068476
Value of the gradient at x0: -0.21474836480000006
Value of x0: 0.13421772800000004
Value of x1: 0.10737418240000003
Value of function f(x0): -0.0073786976294838245
Value of the gradient at x0: -0.17179869184000005
Value of x0: 0.10737418240000003
Value of x1: 0.08589934592000002
Value of function f(x0): -0.004722366482869647
Value of the gradient at x0: -0.13743895347200002
Value of x0: 0.08589934592000002
Value of x1: 0.06871947673600001
Value of function f(x0): -0.003022314549036574
Value of the gradient at x0: -0.10995116277760002
Value of x0: 0.06871947673600001
Value of x1: 0.05497558138880001
Value of function f(x0): -0.0019342813113834073
Value of the gradient at x0: -0.08796093022208001
Value of x0: 0.05497558138880001
Value of x1: 0.04398046511104001
Value of function f(x0): -0.0012379400392853804
Value of the gradient at x0: -0.070368744177664
Value of x0: 0.04398046511104001
Value of x1: 0.035184372088832
Value of function f(x0): -0.0007922816251426435
Value of the gradient at x0: -0.056294995342131206
Value of x0: 0.035184372088832
Value of x1: 0.028147497671065603
Value of function f(x0): -0.0005070602400912918
Value of the gradient at x0: -0.04503599627370496
Value of x0: 0.028147497671065603
Value of x1: 0.02251799813685248
Value of function f(x0): -0.00032451855365842676
Value of the gradient at x0: -0.03602879701896397
Value of x0: 0.02251799813685248
Value of x1: 0.018014398509481985
Value of function f(x0): -0.00020769187434139315
Value of the gradient at x0: -0.028823037615171177
Value of x0: 0.018014398509481985
Value of x1: 0.014411518807585589
Value of function f(x0): -0.0001329227995784916
Value of the gradient at x0: -0.02305843009213694
Value of x0: 0.014411518807585589
Value of x1: 0.01152921504606847
Value of function f(x0): -8.507059173023463e-05
Value of the gradient at x0: -0.018446744073709553
Value of x0: 0.01152921504606847
Value of x1: 0.009223372036854777
Value of function f(x0): -5.444517870735016e-05
Value of the gradient at x0: -0.014757395258967642
Value of x0: 0.009223372036854777
Value of x1: 0.007378697629483821
Value of function f(x0): -3.4844914372704106e-05
Value of the gradient at x0: -0.011805916207174114
Value of x0: 0.007378697629483821
Value of x1: 0.005902958103587057
Value of function f(x0): -2.230074519853063e-05
Value of the gradient at x0: -0.009444732965739291
Value of x0: 0.005902958103587057
Value of x1: 0.004722366482869646
Value of function f(x0): -1.4272476927059604e-05
Value of the gradient at x0: -0.0075557863725914335
Value of x0: 0.004722366482869646
Value of x1: 0.0037778931862957168
Value of function f(x0): -9.134385233318147e-06
Value of the gradient at x0: -0.006044629098073147
Value of x0: 0.0037778931862957168
Value of x1: 0.0030223145490365735
Value of function f(x0): -5.846006549323614e-06
Value of the gradient at x0: -0.004835703278458518
Value of x0: 0.0030223145490365735
Value of x1: 0.002417851639229259
Value of function f(x0): -3.741444191567113e-06
Value of the gradient at x0: -0.003868562622766814
Value of x0: 0.002417851639229259
Value of x1: 0.001934281311383407
Value of function f(x0): -2.3945242826029522e-06
Value of the gradient at x0: -0.0030948500982134514
Value of x0: 0.001934281311383407
Value of x1: 0.0015474250491067257
Value of function f(x0): -1.5324955408658897e-06
Value of the gradient at x0: -0.002475880078570761
Value of x0: 0.0015474250491067257
Value of x1: 0.0012379400392853806
Value of function f(x0): -9.807971461541695e-07
Value of the gradient at x0: -0.001980704062856609
Value of x0: 0.0012379400392853806
Value of x1: 0.0009903520314283045
Value of function f(x0): -6.277101735386685e-07
Value of the gradient at x0: -0.0015845632502852873
Value of x0: 0.0009903520314283045
Value of x1: 0.0007922816251426436


Total number of iterations: 32
Solution value: 0.0007922816251426436
In [9]:
converge(1,0.2)
Value of function f(x0): -0.36
Value of the gradient at x0: -1.2
Value of x0: 1
Value of x1: 0.6
Value of function f(x0): -0.1296
Value of the gradient at x0: -0.72
Value of x0: 0.6
Value of x1: 0.36
Value of function f(x0): -0.046655999999999996
Value of the gradient at x0: -0.432
Value of x0: 0.36
Value of x1: 0.216
Value of function f(x0): -0.016796159999999997
Value of the gradient at x0: -0.2592
Value of x0: 0.216
Value of x1: 0.1296
Value of function f(x0): -0.006046617599999999
Value of the gradient at x0: -0.15552
Value of x0: 0.1296
Value of x1: 0.07776
Value of function f(x0): -0.0021767823359999995
Value of the gradient at x0: -0.09331199999999999
Value of x0: 0.07776
Value of x1: 0.046655999999999996
Value of function f(x0): -0.0007836416409599999
Value of the gradient at x0: -0.055987199999999994
Value of x0: 0.046655999999999996
Value of x1: 0.027993599999999997
Value of function f(x0): -0.0002821109907455999
Value of the gradient at x0: -0.033592319999999995
Value of x0: 0.027993599999999997
Value of x1: 0.016796159999999997
Value of function f(x0): -0.00010155995666841595
Value of the gradient at x0: -0.020155391999999994
Value of x0: 0.016796159999999997
Value of x1: 0.010077695999999997
Value of function f(x0): -3.656158440062974e-05
Value of the gradient at x0: -0.012093235199999997
Value of x0: 0.010077695999999997
Value of x1: 0.006046617599999998
Value of function f(x0): -1.3162170384226707e-05
Value of the gradient at x0: -0.007255941119999998
Value of x0: 0.006046617599999998
Value of x1: 0.003627970559999999
Value of function f(x0): -4.7383813383216126e-06
Value of the gradient at x0: -0.004353564671999998
Value of x0: 0.003627970559999999
Value of x1: 0.002176782335999999
Value of function f(x0): -1.7058172817957804e-06
Value of the gradient at x0: -0.0026121388031999987
Value of x0: 0.002176782335999999
Value of x1: 0.0013060694015999993
Value of function f(x0): -6.140942214464809e-07
Value of the gradient at x0: -0.0015672832819199991
Value of x0: 0.0013060694015999993
Value of x1: 0.0007836416409599996


Total number of iterations: 14
Solution value: 0.0007836416409599996
In [10]:
converge(2,0.4)
Value of function f(x0): -0.15999999999999992
Value of the gradient at x0: -0.7999999999999998
Value of x0: 2
Value of x1: 0.3999999999999999
Value of function f(x0): -0.006399999999999993
Value of the gradient at x0: -0.15999999999999992
Value of x0: 0.3999999999999999
Value of x1: 0.07999999999999996
Value of function f(x0): -0.00025599999999999955
Value of the gradient at x0: -0.03199999999999997
Value of x0: 0.07999999999999996
Value of x1: 0.015999999999999986
Value of function f(x0): -1.0239999999999976e-05
Value of the gradient at x0: -0.0063999999999999925
Value of x0: 0.015999999999999986
Value of x1: 0.0031999999999999963


Total number of iterations: 4
Solution value: 0.0031999999999999963

Based on the above results, we can infer the following:

  • When we kept the x0=0.1 and R=0.1, it took 22 iterations to converge to the maximum value of the function.
  • When we kept the x0=1 and R=0.1, it took 32 iterations to converge to the maximum value of the function.
  • When we kept the x0=1 and R=0.2, it took 14 iterations to converge to the maximum value of the function.
  • When we kept the x0=2 and R=0.4, it took 4 iterations to converge to the maximum value of the function.

With increase in learning rate (R), the number of iterations reduced, i.e., the function reaches maximum value quickly with higher R values. With higher initial value (x0), it takes longer time to converge the function to its maximum value, however, the result is also influenced by the learning rate as discussed above.

If we keep the stopping parameter toa higher value then the number of iteration will reduce as it may reach the value faster while if we keep it very low, it may take longer to reach the solution value.

1.2.4. Now code the algorithm above for 2D case. Do it in matrix form! Show that you get the correct solution if you pick x0 = (2,-3)' and A =[[1,2],[2,8]]

In [10]:
# Function to calculate f(x) value at x
def f(x,A):
    return (-x.T @ A @ x)
In [11]:
# Function to calculate the gradient at x
def grad(x,A):
    return (-2 * (A @ x))
In [21]:
def converge_2d(x0,A,R):
    m0 = []
    l1 = []
    cnt = 0

    while True:
        a = x0
        x1 = x0 + (R) * grad(x0,A)
        m0.append(x0)
        l1.append(x1)
        fx = f(x0,A)
        gradient = grad(x0,A)

        if abs(f(x1,A)-f(x0,A)) < (1e-6) or cnt == 10000:
            print ("\n\nTotal number of iterations:",cnt)
            print ("Solution value:\n",f(a,A))
            break
        else:
            x0 = x1
            print("Value of function f(x0):",fx)
            print("Value of the gradient at x0:\n",gradient)
            print("Value of x0:\n",a)
            print("Value of x1:\n",x1)
            cnt = cnt + 1
In [22]:
x0 = np.matrix("2;-3")
A = np.matrix("1,2;2,8")
R = 0.1 # Learning Rate
print("Value of f(x) at x0:",f(x0,A))
print("Value of gradient at x0:\n",grad(x0,A))
Value of f(x) at x0: [[-52]]
Value of gradient at x0:
 [[ 8]
 [40]]
In [23]:
x0 = np.matrix("2; -3")
A = np.matrix("1,2;2,8")
R = 0.1
converge_2d(x0,A,R)
Value of function f(x0): [[-52]]
Value of the gradient at x0:
 [[ 8]
 [40]]
Value of x0:
 [[ 2]
 [-3]]
Value of x1:
 [[ 2.8]
 [ 1. ]]
Value of function f(x0): [[-27.04]]
Value of the gradient at x0:
 [[ -9.6]
 [-27.2]]
Value of x0:
 [[ 2.8]
 [ 1. ]]
Value of x1:
 [[ 1.84]
 [-1.72]]
Value of function f(x0): [[-14.3936]]
Value of the gradient at x0:
 [[  3.2 ]
 [ 20.16]]
Value of x0:
 [[ 1.84]
 [-1.72]]
Value of x1:
 [[ 2.16 ]
 [ 0.296]]
Value of function f(x0): [[-7.923968]]
Value of the gradient at x0:
 [[ -5.504]
 [-13.376]]
Value of x0:
 [[ 2.16 ]
 [ 0.296]]
Value of x1:
 [[ 1.6096]
 [-1.0416]]
Value of function f(x0): [[-4.5640192]]
Value of the gradient at x0:
 [[  0.9472]
 [ 10.2272]]
Value of x0:
 [[ 1.6096]
 [-1.0416]]
Value of x1:
 [[ 1.70432]
 [-0.01888]]
Value of function f(x0): [[-2.77884805]]
Value of the gradient at x0:
 [[-3.33312]
 [-6.5152 ]]
Value of x0:
 [[ 1.70432]
 [-0.01888]]
Value of x1:
 [[ 1.371008]
 [-0.6704  ]]
Value of function f(x0): [[-1.79865716]]
Value of the gradient at x0:
 [[-0.060416]
 [ 5.242368]]
Value of x0:
 [[ 1.371008]
 [-0.6704  ]]
Value of x1:
 [[ 1.3649664]
 [-0.1461632]]
Value of function f(x0): [[-1.23601129]]
Value of the gradient at x0:
 [[-2.14528  ]
 [-3.1212544]]
Value of x0:
 [[ 1.3649664]
 [-0.1461632]]
Value of x1:
 [[ 1.1504384 ]
 [-0.45828864]]
Value of function f(x0): [[-0.89480493]]
Value of the gradient at x0:
 [[-0.46772224]
 [ 2.73086464]]
Value of x0:
 [[ 1.1504384 ]
 [-0.45828864]]
Value of x1:
 [[ 1.10366618]
 [-0.18520218]]
Value of function f(x0): [[-0.67487229]]
Value of the gradient at x0:
 [[-1.46652365]
 [-1.45142989]]
Value of x0:
 [[ 1.10366618]
 [-0.18520218]]
Value of x1:
 [[ 0.95701381]
 [-0.33034516]]
Value of function f(x0): [[-0.52431932]]
Value of the gradient at x0:
 [[-0.59264696]
 [ 1.45746739]]
Value of x0:
 [[ 0.95701381]
 [-0.33034516]]
Value of x1:
 [[ 0.89774911]
 [-0.18459843]]
Value of function f(x0): [[-0.41567381]]
Value of the gradient at x0:
 [[-1.05710453]
 [-0.63742165]]
Value of x0:
 [[ 0.89774911]
 [-0.18459843]]
Value of x1:
 [[ 0.79203866]
 [-0.24834059]]
Value of function f(x0): [[-0.33392824]]
Value of the gradient at x0:
 [[-0.59071496]
 [ 0.8052948 ]]
Value of x0:
 [[ 0.79203866]
 [-0.24834059]]
Value of x1:
 [[ 0.73296717]
 [-0.16781111]]
Value of function f(x0): [[-0.27052528]]
Value of the gradient at x0:
 [[-0.79468989]
 [-0.2468909 ]]
Value of x0:
 [[ 0.73296717]
 [-0.16781111]]
Value of x1:
 [[ 0.65349818]
 [-0.1925002 ]]
Value of function f(x0): [[-0.22031636]]
Value of the gradient at x0:
 [[-0.53699555]
 [ 0.46601049]]
Value of x0:
 [[ 0.65349818]
 [-0.1925002 ]]
Value of x1:
 [[ 0.59979862]
 [-0.14589915]]
Value of function f(x0): [[-0.18001045]]
Value of the gradient at x0:
 [[-0.61600064]
 [-0.06480807]]
Value of x0:
 [[ 0.59979862]
 [-0.14589915]]
Value of x1:
 [[ 0.53819856]
 [-0.15237996]]
Value of function f(x0): [[-0.14737221]]
Value of the gradient at x0:
 [[-0.46687728]
 [ 0.2852851 ]]
Value of x0:
 [[ 0.53819856]
 [-0.15237996]]
Value of x1:
 [[ 0.49151083]
 [-0.12385145]]
Value of function f(x0): [[-0.12079903]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.09909107]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.08132093]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.06675592]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.05480876]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.04500434]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.03695606]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.03034822]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
 [[ 0.22292471]
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Value of function f(x0): [[-0.02492245]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.22292471]
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Value of x1:
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Value of function f(x0): [[-0.02046701]]
Value of the gradient at x0:
 [[-0.18812078]
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Value of x0:
 [[ 0.20185486]
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Value of x1:
 [[ 0.18304278]
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Value of function f(x0): [[-0.01680821]]
Value of the gradient at x0:
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 [ 0.04228651]]
Value of x0:
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 [-0.0484036 ]]
Value of x1:
 [[ 0.16579567]
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Value of function f(x0): [[-0.01380355]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.01133605]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00930965]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
 [[ 0.12343028]
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Value of function f(x0): [[-0.00764549]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.12343028]
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Value of x1:
 [[ 0.1118352 ]
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Value of function f(x0): [[-0.00627881]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.1118352 ]
 [-0.02973565]]
Value of x1:
 [[ 0.10136242]
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Value of function f(x0): [[-0.00515644]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
 [[ 0.09184701]
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Value of function f(x0): [[-0.0042347]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.09184701]
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Value of x1:
 [[ 0.08324135]
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Value of function f(x0): [[-0.00347773]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.08324135]
 [-0.02209319]]
Value of x1:
 [[ 0.07543035]
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Value of function f(x0): [[-0.00285607]]
Value of the gradient at x0:
 [[-0.07069821]
 [ 0.01892857]]
Value of x0:
 [[ 0.07543035]
 [-0.02004062]]
Value of x1:
 [[ 0.06836053]
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Value of function f(x0): [[-0.00234553]]
Value of the gradient at x0:
 [[-0.06413   ]
 [ 0.01692214]]
Value of x0:
 [[ 0.06836053]
 [-0.01814777]]
Value of x1:
 [[ 0.06194753]
 [-0.01645555]]
Value of function f(x0): [[-0.00192626]]
Value of the gradient at x0:
 [[-0.05807286]
 [ 0.01549871]]
Value of x0:
 [[ 0.06194753]
 [-0.01645555]]
Value of x1:
 [[ 0.05614025]
 [-0.01490568]]
Value of function f(x0): [[-0.00158193]]
Value of the gradient at x0:
 [[-0.05265777]
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Value of x0:
 [[ 0.05614025]
 [-0.01490568]]
Value of x1:
 [[ 0.05087447]
 [-0.01351269]]
Value of function f(x0): [[-0.00129915]]
Value of the gradient at x0:
 [[-0.04769818]
 [ 0.01270516]]
Value of x0:
 [[ 0.05087447]
 [-0.01351269]]
Value of x1:
 [[ 0.04610465]
 [-0.01224217]]
Value of function f(x0): [[-0.00106692]]
Value of the gradient at x0:
 [[-0.04324061]
 [ 0.01145618]]
Value of x0:
 [[ 0.04610465]
 [-0.01224217]]
Value of x1:
 [[ 0.04178059]
 [-0.01109656]]
Value of function f(x0): [[-0.0008762]]
Value of the gradient at x0:
 [[-0.03917496]
 [ 0.01042254]]
Value of x0:
 [[ 0.04178059]
 [-0.01109656]]
Value of x1:
 [[ 0.0378631]
 [-0.0100543]]
Value of function f(x0): [[-0.00071958]]
Value of the gradient at x0:
 [[-0.03550898]
 [ 0.00941646]]
Value of x0:
 [[ 0.0378631]
 [-0.0100543]]
Value of x1:
 [[ 0.0343122 ]
 [-0.00911266]]
Value of function f(x0): [[-0.00059095]]
Value of the gradient at x0:
 [[-0.03217377]
 [ 0.00855372]]
Value of x0:
 [[ 0.0343122 ]
 [-0.00911266]]
Value of x1:
 [[ 0.03109482]
 [-0.00825729]]
Value of function f(x0): [[-0.00048531]]
Value of the gradient at x0:
 [[-0.0291605 ]
 [ 0.00773728]]
Value of x0:
 [[ 0.03109482]
 [-0.00825729]]
Value of x1:
 [[ 0.02817877]
 [-0.00748356]]
Value of function f(x0): [[-0.00039856]]
Value of the gradient at x0:
 [[-0.02642331]
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Value of x0:
 [[ 0.02817877]
 [-0.00748356]]
Value of x1:
 [[ 0.02553644]
 [-0.00678137]]
Value of function f(x0): [[-0.00032732]]
Value of the gradient at x0:
 [[-0.02394738]
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Value of x0:
 [[ 0.02553644]
 [-0.00678137]]
Value of x1:
 [[ 0.0231417 ]
 [-0.00614575]]
Value of function f(x0): [[-0.00026881]]
Value of the gradient at x0:
 [[-0.0217004 ]
 [ 0.00576522]]
Value of x0:
 [[ 0.0231417 ]
 [-0.00614575]]
Value of x1:
 [[ 0.02097166]
 [-0.00556923]]
Value of function f(x0): [[-0.00022076]]
Value of the gradient at x0:
 [[-0.01966641]
 [ 0.00522103]]
Value of x0:
 [[ 0.02097166]
 [-0.00556923]]
Value of x1:
 [[ 0.01900502]
 [-0.00504713]]
Value of function f(x0): [[-0.0001813]]
Value of the gradient at x0:
 [[-0.01782154]
 [ 0.00473395]]
Value of x0:
 [[ 0.01900502]
 [-0.00504713]]
Value of x1:
 [[ 0.01722287]
 [-0.00457373]]
Value of function f(x0): [[-0.00014889]]
Value of the gradient at x0:
 [[-0.01615081]
 [ 0.00428825]]
Value of x0:
 [[ 0.01722287]
 [-0.00457373]]
Value of x1:
 [[ 0.01560779]
 [-0.00414491]]
Value of function f(x0): [[-0.00012227]]
Value of the gradient at x0:
 [[-0.01463594]
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Value of x0:
 [[ 0.01560779]
 [-0.00414491]]
Value of x1:
 [[ 0.01414419]
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Value of function f(x0): [[-0.00010042]]
Value of the gradient at x0:
 [[-0.0132637 ]
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Value of x0:
 [[ 0.01414419]
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Value of x1:
 [[ 0.01281782]
 [-0.00340397]]
Value of function f(x0): [[ -8.24667499e-05]]
Value of the gradient at x0:
 [[-0.01201974]
 [ 0.00319231]]
Value of x0:
 [[ 0.01281782]
 [-0.00340397]]
Value of x1:
 [[ 0.01161585]
 [-0.00308474]]
Value of function f(x0): [[ -6.77254188e-05]]
Value of the gradient at x0:
 [[-0.01089272]
 [ 0.00289251]]
Value of x0:
 [[ 0.01161585]
 [-0.00308474]]
Value of x1:
 [[ 0.01052658]
 [-0.00279549]]
Value of function f(x0): [[ -5.56191721e-05]]
Value of the gradient at x0:
 [[-0.00987118]
 [ 0.00262158]]
Value of x0:
 [[ 0.01052658]
 [-0.00279549]]
Value of x1:
 [[ 0.00953946]
 [-0.00253333]]
Value of function f(x0): [[ -4.56769756e-05]]
Value of the gradient at x0:
 [[-0.00894558]
 [ 0.00237552]]
Value of x0:
 [[ 0.00953946]
 [-0.00253333]]
Value of x1:
 [[ 0.0086449 ]
 [-0.00229578]]
Value of function f(x0): [[ -3.75119949e-05]]
Value of the gradient at x0:
 [[-0.00810667]
 [ 0.00215292]]
Value of x0:
 [[ 0.0086449 ]
 [-0.00229578]]
Value of x1:
 [[ 0.00783423]
 [-0.00208049]]
Value of function f(x0): [[ -3.08065441e-05]]
Value of the gradient at x0:
 [[-0.0073465 ]
 [ 0.00195092]]
Value of x0:
 [[ 0.00783423]
 [-0.00208049]]
Value of x1:
 [[ 0.00709958]
 [-0.0018854 ]]
Value of function f(x0): [[ -2.52997251e-05]]
Value of the gradient at x0:
 [[-0.00665757]
 [ 0.00176805]]
Value of x0:
 [[ 0.00709958]
 [-0.0018854 ]]
Value of x1:
 [[ 0.00643383]
 [-0.00170859]]
Value of function f(x0): [[ -2.07772766e-05]]
Value of the gradient at x0:
 [[-0.00603328]
 [ 0.0016022 ]]
Value of x0:
 [[ 0.00643383]
 [-0.00170859]]
Value of x1:
 [[ 0.0058305 ]
 [-0.00154837]]
Value of function f(x0): [[ -1.70632378e-05]]
Value of the gradient at x0:
 [[-0.0054675 ]
 [ 0.00145199]]
Value of x0:
 [[ 0.0058305 ]
 [-0.00154837]]
Value of x1:
 [[ 0.00528375]
 [-0.00140317]]
Value of function f(x0): [[ -1.40131013e-05]]
Value of the gradient at x0:
 [[-0.0049548 ]
 [ 0.00131581]]
Value of x0:
 [[ 0.00528375]
 [-0.00140317]]
Value of x1:
 [[ 0.00478827]
 [-0.00127159]]
Value of function f(x0): [[ -1.15081916e-05]]
Value of the gradient at x0:
 [[-0.00449016]
 [ 0.00119244]]
Value of x0:
 [[ 0.00478827]
 [-0.00127159]]
Value of x1:
 [[ 0.00433925]
 [-0.00115235]]
Value of function f(x0): [[ -9.45104658e-06]]
Value of the gradient at x0:
 [[-0.0040691]
 [ 0.0010806]]
Value of x0:
 [[ 0.00433925]
 [-0.00115235]]
Value of x1:
 [[ 0.00393234]
 [-0.00104429]]
Value of function f(x0): [[ -7.76162621e-06]]
Value of the gradient at x0:
 [[-0.00368752]
 [ 0.00097928]]
Value of x0:
 [[ 0.00393234]
 [-0.00104429]]
Value of x1:
 [[ 0.00356359]
 [-0.00094636]]
Value of function f(x0): [[ -6.37419792e-06]]
Value of the gradient at x0:
 [[-0.00334173]
 [ 0.00088744]]
Value of x0:
 [[ 0.00356359]
 [-0.00094636]]
Value of x1:
 [[ 0.00322942]
 [-0.00085762]]


Total number of iterations: 68
Solution value:
 [[ -5.23477916e-06]]

As, we can see from the results above, at x0 = (2,-3)' and A =[[1,2],[2,8]], the function converges to its maximum value in 68 iterations. Again the hyperparameters can be updated to change the number of iterations and convergence rate.

As before, experiment with a few different learning rates and see how does this influence the speed of convergence. Comment and explain your findings.

In [25]:
x0 = np.matrix("2; -3")
A = np.matrix("1,2;2,8")
R = 0.001
converge_2d(x0,A,R)
Value of function f(x0): [[-52]]
Value of the gradient at x0:
 [[ 8]
 [40]]
Value of x0:
 [[ 2]
 [-3]]
Value of x1:
 [[ 2.008]
 [-2.96 ]]
Value of function f(x0): [[-50.350144]]
Value of the gradient at x0:
 [[  7.824]
 [ 39.328]]
Value of x0:
 [[ 2.008]
 [-2.96 ]]
Value of x1:
 [[ 2.015824]
 [-2.920672]]
Value of function f(x0): [[-48.755903]]
Value of the gradient at x0:
 [[  7.65104 ]
 [ 38.667456]]
Value of x0:
 [[ 2.015824]
 [-2.920672]]
Value of x1:
 [[ 2.02347504]
 [-2.88200454]]
Value of function f(x0): [[-47.21539573]]
Value of the gradient at x0:
 [[  7.4810681 ]
 [ 38.01817254]]
Value of x0:
 [[ 2.02347504]
 [-2.88200454]]
Value of x1:
 [[ 2.03095611]
 [-2.84398637]]
Value of function f(x0): [[-45.72680459]]
Value of the gradient at x0:
 [[  7.31403327]
 [ 37.37995751]]
Value of x0:
 [[ 2.03095611]
 [-2.84398637]]
Value of x1:
 [[ 2.03827014]
 [-2.80660641]]
Value of function f(x0): [[-44.28837346]]
Value of the gradient at x0:
 [[  7.14988537]
 [ 36.75262206]]
Value of x0:
 [[ 2.03827014]
 [-2.80660641]]
Value of x1:
 [[ 2.04542003]
 [-2.76985379]]
Value of function f(x0): [[-42.89840564]]
Value of the gradient at x0:
 [[  6.98857511]
 [ 36.13598056]]
Value of x0:
 [[ 2.04542003]
 [-2.76985379]]
Value of x1:
 [[ 2.0524086 ]
 [-2.73371781]]
Value of function f(x0): [[-41.55526184]]
Value of the gradient at x0:
 [[  6.83005404]
 [ 35.52985057]]
Value of x0:
 [[ 2.0524086 ]
 [-2.73371781]]
Value of x1:
 [[ 2.05923866]
 [-2.69818796]]
Value of function f(x0): [[-40.25735822]]
Value of the gradient at x0:
 [[  6.67427453]
 [ 34.93405275]]
Value of x0:
 [[ 2.05923866]
 [-2.69818796]]
Value of x1:
 [[ 2.06591293]
 [-2.66325391]]
Value of function f(x0): [[-39.00316452]]
Value of the gradient at x0:
 [[  6.52118977]
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Value of x0:
 [[ 2.06591293]
 [-2.66325391]]
Value of x1:
 [[ 2.07243412]
 [-2.6289055 ]]
Value of function f(x0): [[-37.79120228]]
Value of the gradient at x0:
 [[  6.37075375]
 [ 33.77275147]]
Value of x0:
 [[ 2.07243412]
 [-2.6289055 ]]
Value of x1:
 [[ 2.07880487]
 [-2.59513275]]
Value of function f(x0): [[-36.62004305]]
Value of the gradient at x0:
 [[  6.22292123]
 [ 33.20690444]]
Value of x0:
 [[ 2.07880487]
 [-2.59513275]]
Value of x1:
 [[ 2.0850278 ]
 [-2.56192584]]
Value of function f(x0): [[-35.48830668]]
Value of the gradient at x0:
 [[  6.07764777]
 [ 32.65070228]]
Value of x0:
 [[ 2.0850278 ]
 [-2.56192584]]
Value of x1:
 [[ 2.09110544]
 [-2.52927514]]
Value of function f(x0): [[-34.39465976]]
Value of the gradient at x0:
 [[  5.93488967]
 [ 32.10398045]]
Value of x0:
 [[ 2.09110544]
 [-2.52927514]]
Value of x1:
 [[ 2.09704033]
 [-2.49717116]]
Value of function f(x0): [[-33.33781397]]
Value of the gradient at x0:
 [[  5.79460397]
 [ 31.56657721]]
Value of x0:
 [[ 2.09704033]
 [-2.49717116]]
Value of x1:
 [[ 2.10283494]
 [-2.46560458]]
Value of function f(x0): [[-32.31652457]]
Value of the gradient at x0:
 [[  5.65674845]
 [ 31.03833356]]
Value of x0:
 [[ 2.10283494]
 [-2.46560458]]
Value of x1:
 [[ 2.10849169]
 [-2.43456625]]
Value of function f(x0): [[-31.32958894]]
Value of the gradient at x0:
 [[  5.52128162]
 [ 30.51909322]]
Value of x0:
 [[ 2.10849169]
 [-2.43456625]]
Value of x1:
 [[ 2.11401297]
 [-2.40404715]]
Value of function f(x0): [[-30.37584517]]
Value of the gradient at x0:
 [[  5.38816268]
 [ 30.00870261]]
Value of x0:
 [[ 2.11401297]
 [-2.40404715]]
Value of x1:
 [[ 2.11940113]
 [-2.37403845]]
Value of function f(x0): [[-29.45417061]]
Value of the gradient at x0:
 [[  5.25735155]
 [ 29.50701071]]
Value of x0:
 [[ 2.11940113]
 [-2.37403845]]
Value of x1:
 [[ 2.12465848]
 [-2.34453144]]
Value of function f(x0): [[-28.56348065]]
Value of the gradient at x0:
 [[  5.1288088 ]
 [ 29.01386914]]
Value of x0:
 [[ 2.12465848]
 [-2.34453144]]
Value of x1:
 [[ 2.12978729]
 [-2.31551757]]
Value of function f(x0): [[-27.70272734]]
Value of the gradient at x0:
 [[  5.00249571]
 [ 28.52913199]]
Value of x0:
 [[ 2.12978729]
 [-2.31551757]]
Value of x1:
 [[ 2.13478979]
 [-2.28698844]]
Value of function f(x0): [[-26.87089818]]
Value of the gradient at x0:
 [[  4.87837419]
 [ 28.0526559 ]]
Value of x0:
 [[ 2.13478979]
 [-2.28698844]]
Value of x1:
 [[ 2.13966816]
 [-2.25893578]]
Value of function f(x0): [[-26.06701496]]
Value of the gradient at x0:
 [[  4.75640682]
 [ 27.58429991]]
Value of x0:
 [[ 2.13966816]
 [-2.25893578]]
Value of x1:
 [[ 2.14442457]
 [-2.23135148]]
Value of function f(x0): [[-25.29013254]]
Value of the gradient at x0:
 [[  4.6365568 ]
 [ 27.12392548]]
Value of x0:
 [[ 2.14442457]
 [-2.23135148]]
Value of x1:
 [[ 2.14906112]
 [-2.20422756]]
Value of function f(x0): [[-24.53933775]]
Value of the gradient at x0:
 [[  4.51878799]
 [ 26.67139645]]
Value of x0:
 [[ 2.14906112]
 [-2.20422756]]
Value of x1:
 [[ 2.15357991]
 [-2.17755616]]
Value of function f(x0): [[-23.81374833]]
Value of the gradient at x0:
 [[  4.40306483]
 [ 26.22657895]]
Value of x0:
 [[ 2.15357991]
 [-2.17755616]]
Value of x1:
 [[ 2.15798298]
 [-2.15132958]]
Value of function f(x0): [[-23.11251187]]
Value of the gradient at x0:
 [[  4.28935238]
 [ 25.78934143]]
Value of x0:
 [[ 2.15798298]
 [-2.15132958]]
Value of x1:
 [[ 2.16227233]
 [-2.12554024]]
Value of function f(x0): [[-22.43480479]]
Value of the gradient at x0:
 [[  4.17761631]
 [ 25.35955456]]
Value of x0:
 [[ 2.16227233]
 [-2.12554024]]
Value of x1:
 [[ 2.16644994]
 [-2.10018069]]
Value of function f(x0): [[-21.77983139]]
Value of the gradient at x0:
 [[  4.06782286]
 [ 24.93709122]]
Value of x0:
 [[ 2.16644994]
 [-2.10018069]]
Value of x1:
 [[ 2.17051777]
 [-2.0752436 ]]
Value of function f(x0): [[-21.14682286]]
Value of the gradient at x0:
 [[  3.95993885]
 [ 24.52182647]]
Value of x0:
 [[ 2.17051777]
 [-2.0752436 ]]
Value of x1:
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Value of function f(x0): [[-0.00058545]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00058435]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00058326]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00058216]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00058107]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057998]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057889]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057781]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057673]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057564]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057457]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057349]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057241]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057134]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00057027]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.0005692]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00056813]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00056707]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.000566]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00056494]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00056388]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00056283]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00056177]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00056072]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00055967]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00055862]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00055757]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00055653]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00055548]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00055444]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.0005534]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00055237]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00055133]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.0005503]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00054926]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00054824]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00054721]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00054618]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00054516]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00054414]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00054312]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.0005421]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.0328635 ]
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Value of x1:
 [[ 0.03283268]
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Value of function f(x0): [[-0.00054108]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.03283268]
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Value of x1:
 [[ 0.03280189]
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Value of function f(x0): [[-0.00054007]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.03280189]
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Value of x1:
 [[ 0.03277113]
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Value of function f(x0): [[-0.00053905]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-0.00053804]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.0327404 ]
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Value of x1:
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Value of function f(x0): [[-0.00053704]]
Value of the gradient at x0:
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 [ 0.00814573]]
Value of x0:
 [[ 0.0327097 ]
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Value of x1:
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 [-0.00867839]]
Value of function f(x0): [[-0.00053603]]
Value of the gradient at x0:
 [[-0.0306445 ]
 [ 0.00813809]]
Value of x0:
 [[ 0.03267903]
 [-0.00867839]]
Value of x1:
 [[ 0.03264838]
 [-0.00867025]]
Value of function f(x0): [[-0.00053502]]
Value of the gradient at x0:
 [[-0.03061577]
 [ 0.00813046]]
Value of x0:
 [[ 0.03264838]
 [-0.00867025]]
Value of x1:
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Value of function f(x0): [[-0.00053402]]
Value of the gradient at x0:
 [[-0.03058706]
 [ 0.00812283]]
Value of x0:
 [[ 0.03261777]
 [-0.00866212]]
Value of x1:
 [[ 0.03258718]
 [-0.008654  ]]


Total number of iterations: 4673
Solution value:
 [[-0.00053302]]
In [26]:
x0 = np.matrix("2; -3")
A = np.matrix("1,2;2,8")
R = 0.12
converge_2d(x0,A,R)
Value of function f(x0): [[-52]]
Value of the gradient at x0:
 [[ 8]
 [40]]
Value of x0:
 [[ 2]
 [-3]]
Value of x1:
 [[ 2.96]
 [ 1.8 ]]
Value of function f(x0): [[-55.9936]]
Value of the gradient at x0:
 [[-13.12]
 [-40.64]]
Value of x0:
 [[ 2.96]
 [ 1.8 ]]
Value of x1:
 [[ 1.3856]
 [-3.0768]]
Value of function f(x0): [[-60.60061696]]
Value of the gradient at x0:
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 [ 43.6864]]
Value of x0:
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Value of x1:
 [[ 2.52992 ]
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Value of function f(x0): [[-65.83282849]]
Value of the gradient at x0:
 [[-13.722112]
 [-44.768768]]
Value of x0:
 [[ 2.52992 ]
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Value of x1:
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Value of function f(x0): [[-71.71331868]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
 [[ 2.21049098]
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Value of function f(x0): [[-78.27541879]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-85.56200324]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-93.62507838]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-102.52561557]]
Value of the gradient at x0:
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Value of x0:
 [[ 0.11768666]
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Value of x1:
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Value of function f(x0): [[-112.33359202]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-123.12821198]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-134.99828842]]
Value of the gradient at x0:
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Value of x0:
 [[ 1.72273588]
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Value of x1:
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Value of function f(x0): [[-148.04277117]]
Value of the gradient at x0:
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 [ 68.80553393]]
Value of x0:
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 [-4.18911733]]
Value of x1:
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Value of function f(x0): [[-162.37141257]]
Value of the gradient at x0:
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Value of x0:
 [[ 1.67264155]
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Value of x1:
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Value of function f(x0): [[-178.10556576]]
Value of the gradient at x0:
 [[ 19.54247351]
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-195.37911396]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-214.33953222]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-235.14908546]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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 [-5.39534741]]
Value of function f(x0): [[-257.98616917]]
Value of the gradient at x0:
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Value of x0:
 [[-1.10666096]
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Value of x1:
 [[ 1.74870442]
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Value of function f(x0): [[-283.04680094]]
Value of the gradient at x0:
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 [-94.91348774]]
Value of x0:
 [[ 1.74870442]
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Value of x1:
 [[-1.30854474]
 [-5.89470165]]
Value of function f(x0): [[-310.54627312]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-340.72097899]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-373.83042631]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-410.15945467]]
Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of function f(x0): [[-450.02067448]]
Value of the gradient at x0:
 [[  31.66690642]
 [ 119.80645173]]
Value of x0:
 [[-1.71529349]
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Value of x1:
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Value of function f(x0): [[-493.75714792]]
Value of the gradient at x0:
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 [-125.42205067]]
Value of x0:
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Value of x1:
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Value of function f(x0): [[-541.74533442]]
Value of the gradient at x0:
 [[  34.78799588]
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Value of x0:
 [[-1.92809447]
 [-7.73295173]]
Value of x1:
 [[ 2.24646503]
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Value of function f(x0): [[-594.39832539]]
Value of the gradient at x0:
 [[ -36.65213383]
 [-137.6226752 ]]
Value of x0:
 [[ 2.24646503]
 [ 8.03980094]]
Value of x1:
 [[-2.15179103]
 [-8.47492008]]
Value of function f(x0): [[-652.16939581]]
Value of the gradient at x0:
 [[  38.20326238]
 [ 144.20588543]]
Value of x0:
 [[-2.15179103]
 [-8.47492008]]
Value of x1:
 [[ 2.43260046]
 [ 8.82978617]]
Value of function f(x0): [[-715.55590285]]
Value of the gradient at x0:
 [[ -40.18434559]
 [-151.00698054]]
Value of x0:
 [[ 2.43260046]
 [ 8.82978617]]
Value of x1:
 [[-2.38952101]
 [-9.2910515 ]]
Value of function f(x0): [[-785.10356493]]
Value of the gradient at x0:
 [[  41.94324801]
 [ 158.21490798]]
Value of x0:
 [[-2.38952101]
 [-9.2910515 ]]
Value of x1:
 [[ 2.64366875]
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Value of function f(x0): [[-861.41115791]]
Value of the gradient at x0:
 [[ -44.06628734]
 [-165.69047438]]
Value of x0:
 [[ 2.64366875]
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Value of x1:
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Value of the gradient at x0:
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Value of the gradient at x0:
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Value of x1:
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Value of x0:
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Value of x1:
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Value of the gradient at x0:
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Value of x0:
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Value of x1:
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Value of the gradient at x0:
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Value of x1:
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Value of the gradient at x0:
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C:\Users\Manisha\Anaconda3\lib\site-packages\ipykernel_launcher.py:14: RuntimeWarning: invalid value encountered in subtract
  
C:\Users\Manisha\Anaconda3\lib\site-packages\ipykernel_launcher.py:14: RuntimeWarning: invalid value encountered in less
  
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Value of x1:
 [[  7.31892788e+200]
 [  2.75598946e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -1.24877434e+202]
 [ -4.70234024e+202]]
Value of x0:
 [[  7.31892788e+200]
 [  2.75598946e+201]]
Value of x1:
 [[ -7.66636419e+200]
 [ -2.88681884e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[  1.30805482e+202]
 [  4.92556471e+202]]
Value of x0:
 [[ -7.66636419e+200]
 [ -2.88681884e+201]]
Value of x1:
 [[  8.03029363e+200]
 [  3.02385881e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -1.37014940e+202]
 [ -5.15938584e+202]]
Value of x0:
 [[  8.03029363e+200]
 [  3.02385881e+201]]
Value of x1:
 [[ -8.41149914e+200]
 [ -3.16740420e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[  1.43519166e+202]
 [  5.40430669e+202]]
Value of x0:
 [[ -8.41149914e+200]
 [ -3.16740420e+201]]
Value of x1:
 [[  8.81080082e+200]
 [  3.31776382e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -1.50332155e+202]
 [ -5.66085415e+202]]
Value of x0:
 [[  8.81080082e+200]
 [  3.31776382e+201]]
Value of x1:
 [[ -9.22905773e+200]
 [ -3.47526116e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[  1.57468562e+202]
 [  5.92958016e+202]]
Value of x0:
 [[ -9.22905773e+200]
 [ -3.47526116e+201]]
Value of x1:
 [[  9.66716968e+200]
 [  3.64023504e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -1.64943741e+202]
 [ -6.21106284e+202]]
Value of x0:
 [[  9.66716968e+200]
 [  3.64023504e+201]]
Value of x1:
 [[ -1.01260792e+201]
 [ -3.81304038e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[  1.72773773e+202]
 [  6.50590777e+202]]
Value of x0:
 [[ -1.01260792e+201]
 [ -3.81304038e+201]]
Value of x1:
 [[  1.06067736e+201]
 [  3.99404895e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -1.80975505e+202]
 [ -6.81474926e+202]]
Value of x0:
 [[  1.06067736e+201]
 [  3.99404895e+201]]
Value of x1:
 [[ -1.11102870e+201]
 [ -4.18365017e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[  1.89566581e+202]
 [  7.13825175e+202]]
Value of x0:
 [[ -1.11102870e+201]
 [ -4.18365017e+201]]
Value of x1:
 [[  1.16377027e+201]
 [  4.38225193e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -1.98565483e+202]
 [ -7.47711119e+202]]
Value of x0:
 [[  1.16377027e+201]
 [  4.38225193e+201]]
Value of x1:
 [[ -1.21901552e+201]
 [ -4.59028150e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[  2.07991571e+202]
 [  7.83205661e+202]]
Value of x0:
 [[ -1.21901552e+201]
 [ -4.59028150e+201]]
Value of x1:
 [[  1.27688332e+201]
 [  4.80818643e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -2.17865124e+202]
 [ -8.20385162e+202]]
Value of x0:
 [[  1.27688332e+201]
 [  4.80818643e+201]]
Value of x1:
 [[ -1.33749816e+201]
 [ -5.03643551e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[  2.28207384e+202]
 [  8.59329609e+202]]
Value of x0:
 [[ -1.33749816e+201]
 [ -5.03643551e+201]]
Value of x1:
 [[  1.40099044e+201]
 [  5.27551979e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -2.39040601e+202]
 [ -9.00122784e+202]]
Value of x0:
 [[  1.40099044e+201]
 [  5.27551979e+201]]
Value of x1:
 [[ -1.46749676e+201]
 [ -5.52595362e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[  2.50388080e+202]
 [  9.42852450e+202]]
Value of x0:
 [[ -1.46749676e+201]
 [ -5.52595362e+201]]
Value of x1:
 [[  1.53716020e+201]
 [  5.78827578e+201]]
Value of function f(x0): [[-inf]]
Value of the gradient at x0:
 [[ -2.62274235e+202]
 [ -9.87610532e+202]]
Value of x0:
 [[  1.53716020e+201]
 [  5.78827578e+201]]
Value of x1:
 [[ -1.61013062e+201]
 [ -6.06305061e+201]]


Total number of iterations: 10000
Solution value:
 [[-inf]]

Based on the results above, we observed the following:

  • When we keep R = 0.1, it converges to maxima in 68 iterations.
  • When we keep R = 0.001, it converges to maxima in 4673 iterations.
  • when we keep R = 0.12, it never converges within 10,000 iterations.

Hence, the learning rate near 0.1, converges the function to maximum in less number of steps as compared to other values.

1.2.5. Visualize your algorithm's work:

In [27]:
# Function to calculate gradient Ascent
def gradient_ascent(x0,A,R):
    m0 = []
    l1 = []
    cnt = 0

    while True:
        m0.append(x0[0,0])
        l1.append(x0[1,0])
        x1 = x0 + (R*grad(x0,A))
        m0.append(x1[0,0])
        l1.append(x1[1,0])
        if abs(f(x1,A)-f(x0,A)) < (1e-6) or cnt == 10000:
            break
        else:
            x0 = x1
            cnt = cnt + 1
    return (m0,l1)
In [28]:
## create 30x30 grid matrix
n = 30
ex1 = np.linspace(-5, 4, num=n)
ex2 = np.linspace(-5, 4, num=n)
grid1, grid2 = np.meshgrid(ex1, ex2)
## fill the grid via looping.  You may prefer to use ufuncs instead.
z = np.empty_like(grid1)
for i in range(z.shape[0]):
    for j in range(z.shape[1]):
        x = np.array([grid1[i,j], grid2[i,j]])
        z[i,j] = f(x,A)[0][0]
## Make the plot
plt.figure(figsize=(9,9))
p = plt.contour(grid1, grid2, z,
                levels = [-50, -20, -15, -10, -5, -2, -1, 0, 1, 2, 5, 10, 20, 50],
                colors = 'black')
plt.xlim(-4,6)
plt.ylim(-3.5,2)
plt.clabel(p, inline=1, fontsize=10)

# setting intial value, learning rate and A matrix
x0 = np.matrix("2; -3")
A = np.matrix("1,2;2,8")
R = 0.1

# Getting array for x0 and x1
m0, l1 = gradient_ascent(x0,A,R)

## Adding points on the plot
plt.scatter(m0, l1, c='red', s=30)
plt.plot(m0, l1, c='red')
Out[28]:
[<matplotlib.lines.Line2D at 0x1a4f5cdf550>]

With x0 = (2,-3)' and A =[[1,2],[2,8]], we get proper plot which converges to the maximum value in 68 iterations with learning rate of 0.1. As you ca see in the plot above the starting point is x0 = [[2,3]] which converges to the maximum value of the function at 0 value. The contour plot that is depicts the function values obtained at each point of convergence towards the maximum value of the function. For higher or very low learning rates (as compared to 0.1), the function takes much longer time to converge as the number of iterations are more.

The stopping criteria leveraged for this gradient ascent is abs(f(x1)-f(x0)) < 10^-6 and count of iterations == 10,000. The number of iterations would agin depent on the stopping criteria and the R value selected.

1.2.6. Play with hyperparameters until you achieve convergence for the 5 Dimensional case.

In [29]:
# Function to calculate f(x) value at x
def f(x,A):
    return (-x.T @ A @ x)
In [30]:
# Function to calculate the gradient at x
def grad(x,A):
    return (-2 * (A @ x))
In [43]:
A = np.matrix(np.arange(1,26).reshape(5,5))
A = 1/2 * (A + A.T) + 10 * (np.eye(5))
A.astype(int)
x0 = np.matrix("10; 20; 30; 40; 50")
R = 0.011
f(x0,A)[0]
Out[43]:
matrix([[-437500.]])
In [44]:
converge_2d(x0,A,R)
Value of function f(x0): [[-437500.]]
Value of the gradient at x0:
 [[-2900.]
 [-4000.]
 [-5100.]
 [-6200.]
 [-7300.]]
Value of x0:
 [[10]
 [20]
 [30]
 [40]
 [50]]
Value of x1:
 [[-21.9]
 [-24. ]
 [-26.1]
 [-28.2]
 [-30.3]]
Value of function f(x0): [[-272337.75]]
Value of the gradient at x0:
 [[ 2391.]
 [ 3216.]
 [ 4041.]
 [ 4866.]
 [ 5691.]]
Value of x0:
 [[-21.9]
 [-24. ]
 [-26.1]
 [-28.2]
 [-30.3]]
Value of x1:
 [[  4.401]
 [ 11.376]
 [ 18.351]
 [ 25.326]
 [ 32.301]]
Value of function f(x0): [[-169549.230375]]
Value of the gradient at x0:
 [[-1791.09]
 [-2481.12]
 [-3171.15]
 [-3861.18]
 [-4551.21]]
Value of x0:
 [[  4.401]
 [ 11.376]
 [ 18.351]
 [ 25.326]
 [ 32.301]]
Value of x1:
 [[-15.30099]
 [-15.91632]
 [-16.53165]
 [-17.14698]
 [-17.76231]]
Value of function f(x0): [[-105575.38355518]]
Value of the gradient at x0:
 [[ 1500.1551]
 [ 2008.4112]
 [ 2516.6673]
 [ 3024.9234]
 [ 3533.1795]]
Value of x0:
 [[-15.30099]
 [-15.91632]
 [-16.53165]
 [-17.14698]
 [-17.76231]]
Value of x1:
 [[  1.2007161]
 [  6.1762032]
 [ 11.1516903]
 [ 16.1271774]
 [ 21.1026645]]
Value of function f(x0): [[-65756.14826265]]
Value of the gradient at x0:
 [[-1103.161869]
 [-1537.22232 ]
 [-1971.282771]
 [-2405.343222]
 [-2839.403673]]
Value of x0:
 [[  1.2007161]
 [  6.1762032]
 [ 11.1516903]
 [ 16.1271774]
 [ 21.1026645]]
Value of x1:
 [[-10.93406446]
 [-10.73324232]
 [-10.53242018]
 [-10.33159804]
 [-10.1307759 ]]
Value of function f(x0): [[-40968.9430842]]
Value of the gradient at x0:
 [[  943.90137351]
 [ 1255.85753616]
 [ 1567.81369881]
 [ 1879.76986146]
 [ 2191.72602411]]
Value of x0:
 [[-10.93406446]
 [-10.73324232]
 [-10.53242018]
 [-10.33159804]
 [-10.1307759 ]]
Value of x1:
 [[ -0.55114935]
 [  3.08119058]
 [  6.71353051]
 [ 10.34587043]
 [ 13.97821036]]
Value of function f(x0): [[-25536.94625517]]
Value of the gradient at x0:
 [[ -676.86454409]
 [ -950.91725784]
 [-1224.96997158]
 [-1499.02268532]
 [-1773.07539906]]
Value of x0:
 [[ -0.55114935]
 [  3.08119058]
 [  6.71353051]
 [ 10.34587043]
 [ 13.97821036]]
Value of x1:
 [[-7.99665934]
 [-7.37889926]
 [-6.76113918]
 [-6.1433791 ]
 [-5.52561903]]
Value of function f(x0): [[-15927.51032932]]
Value of the gradient at x0:
 [[  596.14732479]
 [  786.62629869]
 [  977.10527259]
 [ 1167.5842465 ]
 [ 1358.0632204 ]]
Value of x0:
 [[-7.99665934]
 [-7.37889926]
 [-6.76113918]
 [-6.1433791 ]
 [-5.52561903]]
Value of x1:
 [[-1.43903876]
 [ 1.27399003]
 [ 3.98701882]
 [ 6.70004761]
 [ 9.4130764 ]]
Value of function f(x0): [[-9942.2412155]]
Value of the gradient at x0:
 [[ -413.09226934]
 [ -586.96340965]
 [ -760.83454996]
 [ -934.70569027]
 [-1108.57683058]]
Value of x0:
 [[-1.43903876]
 [ 1.27399003]
 [ 3.98701882]
 [ 6.70004761]
 [ 9.4130764 ]]
Value of x1:
 [[-5.98305373]
 [-5.18260748]
 [-4.38216123]
 [-3.58171499]
 [-2.78126874]]
Value of function f(x0): [[-6213.01746658]]
Value of the gradient at x0:
 [[ 378.38558599]
 [ 493.84149804]
 [ 609.29741008]
 [ 724.75332212]
 [ 840.20923417]]
Value of x0:
 [[-5.98305373]
 [-5.18260748]
 [-4.38216123]
 [-3.58171499]
 [-2.78126874]]
Value of x1:
 [[-1.82081228]
 [ 0.249649  ]
 [ 2.32011028]
 [ 4.39057156]
 [ 6.46103284]]
Value of function f(x0): [[-3888.38242811]]
Value of the gradient at x0:
 [[-250.21915064]
 [-361.23168457]
 [-472.2442185 ]
 [-583.25675243]
 [-694.26928637]]
Value of x0:
 [[-1.82081228]
 [ 0.249649  ]
 [ 2.32011028]
 [ 4.39057156]
 [ 6.46103284]]
Value of x1:
 [[-4.57322294]
 [-3.72389953]
 [-2.87457613]
 [-2.02525272]
 [-1.17592931]]
Value of function f(x0): [[-2438.39705149]]
Value of the gradient at x0:
 [[ 241.72538314]
 [ 310.97619878]
 [ 380.22701442]
 [ 449.47783006]
 [ 518.7286457 ]]
Value of x0:
 [[-4.57322294]
 [-3.72389953]
 [-2.87457613]
 [-2.02525272]
 [-1.17592931]]
Value of x1:
 [[-1.91424372]
 [-0.30316134]
 [ 1.30792103]
 [ 2.91900341]
 [ 4.53008579]]
Value of function f(x0): [[-1533.20573761]]
Value of the gradient at x0:
 [[-149.93454057]
 [-221.39381913]
 [-292.85309769]
 [-364.31237626]
 [-435.77165482]]
Value of x0:
 [[-1.91424372]
 [-0.30316134]
 [ 1.30792103]
 [ 2.91900341]
 [ 4.53008579]]
Value of x1:
 [[-3.56352367]
 [-2.73849335]
 [-1.91346304]
 [-1.08843273]
 [-0.26340241]]
Value of function f(x0): [[-967.47080162]]
Value of the gradient at x0:
 [[ 155.71106743]
 [ 196.61435239]
 [ 237.51763735]
 [ 278.42092231]
 [ 319.32420728]]
Value of x0:
 [[-3.56352367]
 [-2.73849335]
 [-1.91346304]
 [-1.08843273]
 [-0.26340241]]
Value of x1:
 [[-1.85070193]
 [-0.57573548]
 [ 0.69923097]
 [ 1.97419742]
 [ 3.24916387]]
Value of function f(x0): [[-613.34923308]]
Value of the gradient at x0:
 [[ -88.43011624]
 [-134.9063743 ]
 [-181.38263235]
 [-227.85889041]
 [-274.33514847]]
Value of x0:
 [[-1.85070193]
 [-0.57573548]
 [ 0.69923097]
 [ 1.97419742]
 [ 3.24916387]]
Value of x1:
 [[-2.82343321]
 [-2.0597056 ]
 [-1.29597799]
 [-0.53225038]
 [ 0.23147723]]
Value of function f(x0): [[-391.23033805]]
Value of the gradient at x0:
 [[ 101.36346656]
 [ 124.96825395]
 [ 148.57304135]
 [ 172.17782874]
 [ 195.78261613]]
Value of x0:
 [[-2.82343321]
 [-2.0597056 ]
 [-1.29597799]
 [-0.53225038]
 [ 0.23147723]]
Value of x1:
 [[-1.70843507]
 [-0.6850548 ]
 [ 0.33832547]
 [ 1.36170574]
 [ 2.38508601]]
Value of function f(x0): [[-251.52544685]]
Value of the gradient at x0:
 [[ -50.9168976 ]
 [ -81.53426708]
 [-112.15163655]
 [-142.76900603]
 [-173.38637551]]
Value of x0:
 [[-1.70843507]
 [-0.6850548 ]
 [ 0.33832547]
 [ 1.36170574]
 [ 2.38508601]]
Value of x1:
 [[-2.26852095]
 [-1.58193174]
 [-0.89534253]
 [-0.20875333]
 [ 0.47783588]]
Value of function f(x0): [[-163.33506252]]
Value of the gradient at x0:
 [[  66.84904388]
 [  79.97753575]
 [  93.10602762]
 [ 106.23451949]
 [ 119.36301136]]
Value of x0:
 [[-2.26852095]
 [-1.58193174]
 [-0.89534253]
 [-0.20875333]
 [ 0.47783588]]
Value of x1:
 [[-1.53318146]
 [-0.70217885]
 [ 0.12882377]
 [ 0.95982639]
 [ 1.790829  ]]
Value of function f(x0): [[-107.3957385]]
Value of the gradient at x0:
 [[ -28.21419168]
 [ -48.69895713]
 [ -69.18372258]
 [ -89.66848804]
 [-110.15325349]]
Value of x0:
 [[-1.53318146]
 [-0.70217885]
 [ 0.12882377]
 [ 0.95982639]
 [ 1.790829  ]]
Value of x1:
 [[-1.84353757]
 [-1.23786738]
 [-0.63219718]
 [-0.02652698]
 [ 0.57914322]]
Value of function f(x0): [[-71.69021891]]
Value of the gradient at x0:
 [[ 44.78434208]
 [ 51.63685348]
 [ 58.48936488]
 [ 65.34187628]
 [ 72.19438768]]
Value of x0:
 [[-1.84353757]
 [-1.23786738]
 [-0.63219718]
 [-0.02652698]
 [ 0.57914322]]
Value of x1:
 [[-1.35090981]
 [-0.66986199]
 [ 0.01118584]
 [ 0.69223366]
 [ 1.37328148]]
Value of function f(x0): [[-48.71496145]]
Value of the gradient at x0:
 [[-14.62768166]
 [-28.58421318]
 [-42.5407447 ]
 [-56.49727621]
 [-70.45380773]]
Value of x0:
 [[-1.35090981]
 [-0.66986199]
 [ 0.01118584]
 [ 0.69223366]
 [ 1.37328148]]
Value of x1:
 [[-1.51181431]
 [-0.98428833]
 [-0.45676236]
 [ 0.07076362]
 [ 0.5982896 ]]
Value of function f(x0): [[-33.77910695]]
Value of the gradient at x0:
 [[ 30.55809253]
 [ 33.71044369]
 [ 36.86279485]
 [ 40.01514602]
 [ 43.16749718]]
Value of x0:
 [[-1.51181431]
 [-0.98428833]
 [-0.45676236]
 [ 0.07076362]
 [ 0.5982896 ]]
Value of x1:
 [[-1.17567529]
 [-0.61347345]
 [-0.05127161]
 [ 0.51093023]
 [ 1.07313206]]
Value of function f(x0): [[-23.94538412]]
Value of the gradient at x0:
 [[ -6.62959164]
 [-16.33548003]
 [-26.04136842]
 [-35.74725682]
 [-45.45314521]]
Value of x0:
 [[-1.17567529]
 [-0.61347345]
 [-0.05127161]
 [ 0.51093023]
 [ 1.07313206]]
Value of x1:
 [[-1.2486008 ]
 [-0.79316373]
 [-0.33772667]
 [ 0.1177104 ]
 [ 0.57314747]]
Value of function f(x0): [[-17.37055731]]
Value of the gradient at x0:
 [[ 21.28665855]
 [ 22.30971718]
 [ 23.33277582]
 [ 24.35583445]
 [ 25.37889308]]
Value of x0:
 [[-1.2486008 ]
 [-0.79316373]
 [-0.33772667]
 [ 0.1177104 ]
 [ 0.57314747]]
Value of x1:
 [[-1.01444755]
 [-0.54775684]
 [-0.08106613]
 [ 0.38562458]
 [ 0.85231529]]
Value of function f(x0): [[-12.89457297]]
Value of the gradient at x0:
 [[ -2.03786241]
 [ -8.93969269]
 [-15.84152298]
 [-22.74335326]
 [-29.64518355]]
Value of x0:
 [[-1.01444755]
 [-0.54775684]
 [-0.08106613]
 [ 0.38562458]
 [ 0.85231529]]
Value of x1:
 [[-1.03686404]
 [-0.64609346]
 [-0.25532288]
 [ 0.13544769]
 [ 0.52621827]]
Value of function f(x0): [[-9.78452884]]
Value of the gradient at x0:
 [[ 15.163648  ]
 [ 15.00792296]
 [ 14.85219792]
 [ 14.69647288]
 [ 14.54074784]]
Value of x0:
 [[-1.03686404]
 [-0.64609346]
 [-0.25532288]
 [ 0.13544769]
 [ 0.52621827]]
Value of x1:
 [[-0.87006391]
 [-0.48100631]
 [-0.09194871]
 [ 0.2971089 ]
 [ 0.6861665 ]]
Value of function f(x0): [[-7.57504054]]
Value of the gradient at x0:
 [[  0.49423157]
 [ -4.52845928]
 [ -9.55115012]
 [-14.57384097]
 [-19.59653182]]
Value of x0:
 [[-0.87006391]
 [-0.48100631]
 [-0.09194871]
 [ 0.2971089 ]
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Value of x1:
 [[-0.02012762]
 [-0.0118135 ]
 [-0.00349938]
 [ 0.00481475]
 [ 0.01312887]]
Value of function f(x0): [[-0.0027763]]
Value of the gradient at x0:
 [[ 0.14866137]
 [ 0.08736018]
 [ 0.02605899]
 [-0.0352422 ]
 [-0.09654339]]
Value of x0:
 [[-0.02012762]
 [-0.0118135 ]
 [-0.00349938]
 [ 0.00481475]
 [ 0.01312887]]
Value of x1:
 [[-0.01849235]
 [-0.01085254]
 [-0.00321273]
 [ 0.00442708]
 [ 0.01206689]]
Value of function f(x0): [[-0.00234389]]
Value of the gradient at x0:
 [[ 0.13634923]
 [ 0.07993484]
 [ 0.02352044]
 [-0.03289395]
 [-0.08930834]]
Value of x0:
 [[-0.01849235]
 [-0.01085254]
 [-0.00321273]
 [ 0.00442708]
 [ 0.01206689]]
Value of x1:
 [[-0.01699251]
 [-0.00997325]
 [-0.002954  ]
 [ 0.00406525]
 [ 0.0110845 ]]
Value of function f(x0): [[-0.00197882]]
Value of the gradient at x0:
 [[ 0.12547516]
 [ 0.07371018]
 [ 0.02194521]
 [-0.02981976]
 [-0.08158474]]
Value of x0:
 [[-0.01699251]
 [-0.00997325]
 [-0.002954  ]
 [ 0.00406525]
 [ 0.0110845 ]]
Value of x1:
 [[-0.01561228]
 [-0.00916244]
 [-0.0027126 ]
 [ 0.00373723]
 [ 0.01018707]]
Value of function f(x0): [[-0.00167062]]
Value of the gradient at x0:
 [[ 0.11513769]
 [ 0.06751909]
 [ 0.01990049]
 [-0.0277181 ]
 [-0.0753367 ]]
Value of x0:
 [[-0.01561228]
 [-0.00916244]
 [-0.0027126 ]
 [ 0.00373723]
 [ 0.01018707]]
Value of x1:
 [[-0.01434576]
 [-0.00841973]
 [-0.0024937 ]
 [ 0.00343233]
 [ 0.00935837]]
Value of function f(x0): [[-0.00141041]]
Value of the gradient at x0:
 [[ 0.10591229]
 [ 0.06220262]
 [ 0.01849295]
 [-0.02521672]
 [-0.06892639]]
Value of x0:
 [[-0.01434576]
 [-0.00841973]
 [-0.0024937 ]
 [ 0.00343233]
 [ 0.00935837]]
Value of x1:
 [[-0.01318073]
 [-0.0077355 ]
 [-0.00229028]
 [ 0.00315495]
 [ 0.00860018]]
Value of function f(x0): [[-0.00119074]]
Value of the gradient at x0:
 [[ 0.0972204 ]
 [ 0.05702418]
 [ 0.01682796]
 [-0.02336825]
 [-0.06356447]]
Value of x0:
 [[-0.01318073]
 [-0.0077355 ]
 [-0.00229028]
 [ 0.00315495]
 [ 0.00860018]]
Value of x1:
 [[-0.01211131]
 [-0.00710824]
 [-0.00210517]
 [ 0.0028979 ]
 [ 0.00790097]]
Value of function f(x0): [[-0.00100528]]
Value of the gradient at x0:
 [[ 0.08940388]
 [ 0.05249761]
 [ 0.01559133]
 [-0.02131495]
 [-0.05822123]]
Value of x0:
 [[-0.01211131]
 [-0.00710824]
 [-0.00210517]
 [ 0.0028979 ]
 [ 0.00790097]]
Value of x1:
 [[-0.01112786]
 [-0.00653076]
 [-0.00193366]
 [ 0.00266343]
 [ 0.00726053]]
Value of function f(x0): [[-0.0008487]]
Value of the gradient at x0:
 [[ 0.08208785]
 [ 0.04815581]
 [ 0.01422378]
 [-0.01970825]
 [-0.05364029]]
Value of x0:
 [[-0.01112786]
 [-0.00653076]
 [-0.00193366]
 [ 0.00266343]
 [ 0.00726053]]
Value of x1:
 [[-0.0102249 ]
 [-0.00600105]
 [-0.0017772 ]
 [ 0.00244664]
 [ 0.00667049]]
Value of function f(x0): [[-0.00071652]]
Value of the gradient at x0:
 [[ 0.07547135]
 [ 0.04431052]
 [ 0.01314969]
 [-0.01801115]
 [-0.04917198]]
Value of x0:
 [[-0.0102249 ]
 [-0.00600105]
 [-0.0017772 ]
 [ 0.00244664]
 [ 0.00667049]]
Value of x1:
 [[-0.00939471]
 [-0.00551363]
 [-0.00163256]
 [ 0.00224852]
 [ 0.0061296 ]]
Value of function f(x0): [[-0.00060492]]
Value of the gradient at x0:
 [[ 0.06930855]
 [ 0.0406637 ]
 [ 0.01201885]
 [-0.01662599]
 [-0.04527084]]
Value of x0:
 [[-0.00939471]
 [-0.00551363]
 [-0.00163256]
 [ 0.00224852]
 [ 0.0061296 ]]
Value of x1:
 [[-0.00863232]
 [-0.00506633]
 [-0.00150035]
 [ 0.00206563]
 [ 0.00563162]]
Value of function f(x0): [[-0.0005107]]
Value of the gradient at x0:
 [[ 0.06371175]
 [ 0.03740255]
 [ 0.01109334]
 [-0.01521586]
 [-0.04152506]]
Value of x0:
 [[-0.00863232]
 [-0.00506633]
 [-0.00150035]
 [ 0.00206563]
 [ 0.00563162]]
Value of x1:
 [[-0.00793149]
 [-0.00465491]
 [-0.00137832]
 [ 0.00189826]
 [ 0.00517484]]
Value of function f(x0): [[-0.00043116]]
Value of the gradient at x0:
 [[ 0.05851736]
 [ 0.03433538]
 [ 0.0101534 ]
 [-0.01402858]
 [-0.03821056]]
Value of x0:
 [[-0.00793149]
 [-0.00465491]
 [-0.00137832]
 [ 0.00189826]
 [ 0.00517484]]
Value of x1:
 [[-0.0072878 ]
 [-0.00427722]
 [-0.00126664]
 [ 0.00174395]
 [ 0.00475453]]
Value of function f(x0): [[-0.000364]]
Value of the gradient at x0:
 [[ 0.05378553]
 [ 0.03157297]
 [ 0.0093604 ]
 [-0.01285217]
 [-0.03506474]]
Value of x0:
 [[-0.0072878 ]
 [-0.00427722]
 [-0.00126664]
 [ 0.00174395]
 [ 0.00475453]]
Value of x1:
 [[-0.00669616]
 [-0.00392991]
 [-0.00116367]
 [ 0.00160257]
 [ 0.00436881]]
Value of function f(x0): [[-0.00030731]]
Value of the gradient at x0:
 [[ 0.0494055 ]
 [ 0.02899077]
 [ 0.00857604]
 [-0.0118387 ]
 [-0.03225343]]
Value of x0:
 [[-0.00669616]
 [-0.00392991]
 [-0.00116367]
 [ 0.00160257]
 [ 0.00436881]]
Value of x1:
 [[-0.0061527 ]
 [-0.00361101]
 [-0.00106933]
 [ 0.00147235]
 [ 0.00401403]]
Value of function f(x0): [[-0.00025945]]
Value of the gradient at x0:
 [[ 0.04540647]
 [ 0.02665289]
 [ 0.0078993 ]
 [-0.01085428]
 [-0.02960787]]
Value of x0:
 [[-0.0061527 ]
 [-0.00361101]
 [-0.00106933]
 [ 0.00147235]
 [ 0.00401403]]
Value of x1:
 [[-0.00565322]
 [-0.00331783]
 [-0.00098244]
 [ 0.00135295]
 [ 0.00368834]]
Value of function f(x0): [[-0.00021904]]
Value of the gradient at x0:
 [[ 0.04171195]
 [ 0.02447739]
 [ 0.00724282]
 [-0.00999174]
 [-0.02722631]]
Value of x0:
 [[-0.00565322]
 [-0.00331783]
 [-0.00098244]
 [ 0.00135295]
 [ 0.00368834]]
Value of x1:
 [[-0.00519439]
 [-0.00304858]
 [-0.00090277]
 [ 0.00124304]
 [ 0.00338885]]
Value of function f(x0): [[-0.00018492]]
Value of the gradient at x0:
 [[ 0.03833316]
 [ 0.02250007]
 [ 0.00666698]
 [-0.00916611]
 [-0.0249992 ]]
Value of x0:
 [[-0.00519439]
 [-0.00304858]
 [-0.00090277]
 [ 0.00124304]
 [ 0.00338885]]
Value of x1:
 [[-0.00477273]
 [-0.00280108]
 [-0.00082943]
 [ 0.00114221]
 [ 0.00311386]]
Value of function f(x0): [[-0.00015612]]
Value of the gradient at x0:
 [[ 0.03521613]
 [ 0.02066622]
 [ 0.0061163 ]
 [-0.00843361]
 [-0.02298353]]
Value of x0:
 [[-0.00477273]
 [-0.00280108]
 [-0.00082943]
 [ 0.00114221]
 [ 0.00311386]]
Value of x1:
 [[-0.00438535]
 [-0.00257375]
 [-0.00076215]
 [ 0.00104944]
 [ 0.00286104]]
Value of function f(x0): [[-0.0001318]]
Value of the gradient at x0:
 [[ 0.03236197]
 [ 0.01899466]
 [ 0.00562735]
 [-0.00773997]
 [-0.02110728]]
Value of x0:
 [[-0.00438535]
 [-0.00257375]
 [-0.00076215]
 [ 0.00104944]
 [ 0.00286104]]
Value of x1:
 [[-0.00402937]
 [-0.00236481]
 [-0.00070025]
 [ 0.0009643 ]
 [ 0.00262886]]
Value of function f(x0): [[-0.00011127]]
Value of the gradient at x0:
 [[ 0.02973171]
 [ 0.01744818]
 [ 0.00516465]
 [-0.00711888]
 [-0.01940241]]
Value of x0:
 [[-0.00402937]
 [-0.00236481]
 [-0.00070025]
 [ 0.0009643 ]
 [ 0.00262886]]
Value of x1:
 [[-0.00370232]
 [-0.00217288]
 [-0.00064344]
 [ 0.000886  ]
 [ 0.00241543]]
Value of function f(x0): [[ -9.39433563e-05]]
Value of the gradient at x0:
 [[ 0.02732108]
 [ 0.01603559]
 [ 0.00475011]
 [-0.00653538]
 [-0.01782087]]
Value of x0:
 [[-0.00370232]
 [-0.00217288]
 [-0.00064344]
 [ 0.000886  ]
 [ 0.00241543]]
Value of x1:
 [[-0.00340179]
 [-0.00199649]
 [-0.00059119]
 [ 0.00081411]
 [ 0.0022194 ]]
Value of function f(x0): [[ -7.93114765e-05]]
Value of the gradient at x0:
 [[ 0.02510128]
 [ 0.01473107]
 [ 0.00436085]
 [-0.00600936]
 [-0.01637958]]
Value of x0:
 [[-0.00340179]
 [-0.00199649]
 [-0.00059119]
 [ 0.00081411]
 [ 0.0022194 ]]
Value of x1:
 [[-0.00312567]
 [-0.00183445]
 [-0.00054322]
 [ 0.000748  ]
 [ 0.00203923]]
Value of function f(x0): [[ -6.69585434e-05]]
Value of the gradient at x0:
 [[ 0.02306549]
 [ 0.01353764]
 [ 0.00400979]
 [-0.00551806]
 [-0.01504591]]
Value of x0:
 [[-0.00312567]
 [-0.00183445]
 [-0.00054322]
 [ 0.000748  ]
 [ 0.00203923]]
Value of x1:
 [[-0.00287195]
 [-0.00168553]
 [-0.00049911]
 [ 0.0006873 ]
 [ 0.00187372]]
Value of function f(x0): [[ -5.65296063e-05]]
Value of the gradient at x0:
 [[ 0.02119192]
 [ 0.01243697]
 [ 0.00368202]
 [-0.00507293]
 [-0.01382789]]
Value of x0:
 [[-0.00287195]
 [-0.00168553]
 [-0.00049911]
 [ 0.0006873 ]
 [ 0.00187372]]
Value of x1:
 [[-0.00263884]
 [-0.00154873]
 [-0.00045861]
 [ 0.0006315 ]
 [ 0.00172162]]
Value of function f(x0): [[ -4.77249985e-05]]
Value of the gradient at x0:
 [[ 0.01947281]
 [ 0.01142889]
 [ 0.00338496]
 [-0.00465897]
 [-0.0127029 ]]
Value of x0:
 [[-0.00263884]
 [-0.00154873]
 [-0.00045861]
 [ 0.0006315 ]
 [ 0.00172162]]
Value of x1:
 [[-0.00242464]
 [-0.00142301]
 [-0.00042138]
 [ 0.00058025]
 [ 0.00158189]]
Value of function f(x0): [[ -4.02917273e-05]]
Value of the gradient at x0:
 [[ 0.01789137]
 [ 0.01050007]
 [ 0.00310877]
 [-0.00428253]
 [-0.01167383]]
Value of x0:
 [[-0.00242464]
 [-0.00142301]
 [-0.00042138]
 [ 0.00058025]
 [ 0.00158189]]
Value of x1:
 [[-0.00222784]
 [-0.00130751]
 [-0.00038718]
 [ 0.00053315]
 [ 0.00145347]]
Value of function f(x0): [[ -3.40162041e-05]]
Value of the gradient at x0:
 [[ 0.01643977]
 [ 0.00964867]
 [ 0.00285756]
 [-0.00393355]
 [-0.01072466]]
Value of x0:
 [[-0.00222784]
 [-0.00130751]
 [-0.00038718]
 [ 0.00053315]
 [ 0.00145347]]
Value of x1:
 [[-0.002047  ]
 [-0.00120137]
 [-0.00035575]
 [ 0.00048988]
 [ 0.0013355 ]]
Value of function f(x0): [[ -2.87181072e-05]]
Value of the gradient at x0:
 [[ 0.01510484]
 [ 0.00886478]
 [ 0.00262472]
 [-0.00361534]
 [-0.0098554 ]]
Value of x0:
 [[-0.002047  ]
 [-0.00120137]
 [-0.00035575]
 [ 0.00048988]
 [ 0.0013355 ]]
Value of x1:
 [[-0.00188085]
 [-0.00110386]
 [-0.00032688]
 [ 0.00045011]
 [ 0.00122709]]
Value of function f(x0): [[ -2.42452002e-05]]
Value of the gradient at x0:
 [[ 0.01387918]
 [ 0.00814577]
 [ 0.00241237]
 [-0.00332103]
 [-0.00905444]]
Value of x0:
 [[-0.00188085]
 [-0.00110386]
 [-0.00032688]
 [ 0.00045011]
 [ 0.00122709]]
Value of x1:
 [[-0.00172817]
 [-0.00101426]
 [-0.00030034]
 [ 0.00041358]
 [ 0.00112749]]
Value of function f(x0): [[ -2.04689581e-05]]
Value of the gradient at x0:
 [[ 0.01275228]
 [ 0.00748414]
 [ 0.00221601]
 [-0.00305213]
 [-0.00832027]]
Value of x0:
 [[-0.00172817]
 [-0.00101426]
 [-0.00030034]
 [ 0.00041358]
 [ 0.00112749]]
Value of x1:
 [[-0.0015879 ]
 [-0.00093193]
 [-0.00027596]
 [ 0.00038   ]
 [ 0.00103597]]
Value of function f(x0): [[ -1.72808739e-05]]
Value of the gradient at x0:
 [[ 0.01171742]
 [ 0.006877  ]
 [ 0.00203657]
 [-0.00280386]
 [-0.00764429]]
Value of x0:
 [[-0.0015879 ]
 [-0.00093193]
 [-0.00027596]
 [ 0.00038   ]
 [ 0.00103597]]
Value of x1:
 [[-0.00145901]
 [-0.00085629]
 [-0.00025356]
 [ 0.00034916]
 [ 0.00095188]]
Value of function f(x0): [[ -1.45893406e-05]]
Value of the gradient at x0:
 [[ 0.01076612]
 [ 0.00631852]
 [ 0.00187091]
 [-0.00257669]
 [-0.00702429]]
Value of x0:
 [[-0.00145901]
 [-0.00085629]
 [-0.00025356]
 [ 0.00034916]
 [ 0.00095188]]
Value of x1:
 [[-0.00134058]
 [-0.00078678]
 [-0.00023298]
 [ 0.00032082]
 [ 0.00087462]]
Value of function f(x0): [[ -1.23170194e-05]]
Value of the gradient at x0:
 [[ 0.00989238]
 [ 0.00580585]
 [ 0.00171932]
 [-0.00236721]
 [-0.00645374]]
Value of x0:
 [[-0.00134058]
 [-0.00078678]
 [-0.00023298]
 [ 0.00032082]
 [ 0.00087462]]
Value of x1:
 [[-0.00123176]
 [-0.00072292]
 [-0.00021407]
 [ 0.00029478]
 [ 0.00080363]]
Value of function f(x0): [[ -1.03986171e-05]]
Value of the gradient at x0:
 [[ 0.00908929]
 [ 0.00533442]
 [ 0.00157955]
 [-0.00217532]
 [-0.00593019]]
Value of x0:
 [[-0.00123176]
 [-0.00072292]
 [-0.00021407]
 [ 0.00029478]
 [ 0.00080363]]
Value of x1:
 [[-0.00113178]
 [-0.00066424]
 [-0.00019669]
 [ 0.00027085]
 [ 0.00073839]]
Value of function f(x0): [[ -8.77901016e-06]]
Value of the gradient at x0:
 [[ 0.00835161]
 [ 0.00490156]
 [ 0.00145151]
 [-0.00199854]
 [-0.00544859]]
Value of x0:
 [[-0.00113178]
 [-0.00066424]
 [-0.00019669]
 [ 0.00027085]
 [ 0.00073839]]
Value of x1:
 [[-0.00103991]
 [-0.00061032]
 [-0.00018073]
 [ 0.00024887]
 [ 0.00067846]]
Value of function f(x0): [[ -7.41166045e-06]]
Value of the gradient at x0:
 [[ 0.00767363]
 [ 0.00450359]
 [ 0.00133355]
 [-0.00183648]
 [-0.00500652]]
Value of x0:
 [[-0.00103991]
 [-0.00061032]
 [-0.00018073]
 [ 0.00024887]
 [ 0.00067846]]
Value of x1:
 [[-0.0009555 ]
 [-0.00056078]
 [-0.00016606]
 [ 0.00022866]
 [ 0.00062339]]


Total number of iterations: 107
Solution value:
 [[ -6.25727839e-06]]

1.2.6. Show and comment your results. In particular I'd like to see your comments comparing the easiness and speed of getting the 1D, 2D and 5D case to converge.

We have following observations from the convergence plots above;

  • In 1 dimensional, we received convergence in less than 40 iterations for R values around 0.1.
  • In 2 dimensional, we recieved convergence in slightly higher number of steps ~68 as compared to the 1 dimension.
  • In 5 dimensional, we received convergence in 107 steps for an R value of 0.11. This was almost double the number of steps taken for 2 Dimensional.

Hence, the convergence rate is faster in 1D as compared to 2D and 5D. It was easier to achieve convergence in 1D as compared to the other two and again easier for 2D as compared to 5D.

1.3.1. Compute the condition number for matrix A in the example above.

In [45]:
ev = np.linalg.eig(A)
print('\neigenvalues\n', ev)
val, vec = np.linalg.eig(A)
print('\neigenvalues\n', val)
print('\ncondition number', np.linalg.cond(A))
print('\neigenvectors\n', vec)
eigenvalues
 (array([ 81.31043674,   3.68956326,  10.        ,  10.        ,  10.        ]), matrix([[ -2.48318441e-01,  -7.33715171e-01,   6.32455532e-01,
           1.22417347e-01,   2.57430627e-01],
        [ -3.38483952e-01,  -4.30614229e-01,  -6.32455532e-01,
           3.60186977e-01,  -4.77461925e-01],
        [ -4.28649462e-01,  -1.27513288e-01,  -3.16227766e-01,
          -5.28528454e-01,   4.99558849e-01],
        [ -5.18814973e-01,   1.75587654e-01,   3.22700303e-16,
          -5.13173410e-01,  -5.96454433e-01],
        [ -6.08980483e-01,   4.78688595e-01,   3.16227766e-01,
           5.59097540e-01,   3.16926881e-01]]))

eigenvalues
 [ 81.31043674   3.68956326  10.          10.          10.        ]

condition number 22.0379570765

eigenvectors
 [[ -2.48318441e-01  -7.33715171e-01   6.32455532e-01   1.22417347e-01
    2.57430627e-01]
 [ -3.38483952e-01  -4.30614229e-01  -6.32455532e-01   3.60186977e-01
   -4.77461925e-01]
 [ -4.28649462e-01  -1.27513288e-01  -3.16227766e-01  -5.28528454e-01
    4.99558849e-01]
 [ -5.18814973e-01   1.75587654e-01   3.22700303e-16  -5.13173410e-01
   -5.96454433e-01]
 [ -6.08980483e-01   4.78688595e-01   3.16227766e-01   5.59097540e-01
    3.16926881e-01]]
In [46]:
condition_number = np.sqrt(max(abs(val))/(min(abs(val))))
print("Condition number using python formula:",np.linalg.cond(A))
print("Condition number using Greene's method:",condition_number)
Condition number using python formula: 22.0379570765
Condition number using Greene's method: 4.69446025401

1.3.2. Show, on computer, that it is singular! Show it without error messages but using relevant linear algebra functions/properties instead.

In [47]:
A = np.ones(4).reshape(2,2).astype(int)
A
Out[47]:
array([[1, 1],
       [1, 1]])
In [48]:
Determinant_A = np.linalg.det(A)
Determinant_A
Out[48]:
0.0

Since, the determinant of A matrix is zero, we can conclude that A is a singular matrix.

1.3.3. Run your algorithm for each of these three cases. For each alpha:

  • (a) report the hyperparameters
  • (b) compute the corresponding C
  • (c) print it's eigenvalues and condition number. Please state which definition of eigenvalues you are using if not following Greene!
  • (d) solve the problem using GA
  • (e) report the location of maximum and the number of iterations it took
  • (f) visualize the process using a gure, similar to the 8 above. Try to play with the 'level' argument of the contour plot to make the shape of the resulting object clear.
In [50]:
alpha = 1
x0 = np.matrix("2; -3")
C = np.matrix(A + np.eye(2) * alpha)
C
Out[50]:
matrix([[ 2.,  1.],
        [ 1.,  2.]])
In [51]:
def eig_cond(A):
    ev = np.linalg.eig(A)
    val, vec = np.linalg.eig(A)
    condn = np.sqrt(max(abs(val))/abs(min(abs(val))))
    return (val, vec, condn)
In [52]:
val, vec, condn = eig_cond(C)
print("eigen values:",val)
print("eigen vectors:\n",vec)
print("condition number:",condn)
eigen values: [ 3.  1.]
eigen vectors:
 [[ 0.70710678 -0.70710678]
 [ 0.70710678  0.70710678]]
condition number: 1.73205080757
In [53]:
converge_2d(x0,C,0.25) # Hyperparameter R = 0.25
Value of function f(x0): [[-14.]]
Value of the gradient at x0:
 [[-2.]
 [ 8.]]
Value of x0:
 [[ 2]
 [-3]]
Value of x1:
 [[ 1.5]
 [-1. ]]
Value of function f(x0): [[-3.5]]
Value of the gradient at x0:
 [[-4.]
 [ 1.]]
Value of x0:
 [[ 1.5]
 [-1. ]]
Value of x1:
 [[ 0.5 ]
 [-0.75]]
Value of function f(x0): [[-0.875]]
Value of the gradient at x0:
 [[-0.5]
 [ 2. ]]
Value of x0:
 [[ 0.5 ]
 [-0.75]]
Value of x1:
 [[ 0.375]
 [-0.25 ]]
Value of function f(x0): [[-0.21875]]
Value of the gradient at x0:
 [[-1.  ]
 [ 0.25]]
Value of x0:
 [[ 0.375]
 [-0.25 ]]
Value of x1:
 [[ 0.125 ]
 [-0.1875]]
Value of function f(x0): [[-0.0546875]]
Value of the gradient at x0:
 [[-0.125]
 [ 0.5  ]]
Value of x0:
 [[ 0.125 ]
 [-0.1875]]
Value of x1:
 [[ 0.09375]
 [-0.0625 ]]
Value of function f(x0): [[-0.01367188]]
Value of the gradient at x0:
 [[-0.25  ]
 [ 0.0625]]
Value of x0:
 [[ 0.09375]
 [-0.0625 ]]
Value of x1:
 [[ 0.03125 ]
 [-0.046875]]
Value of function f(x0): [[-0.00341797]]
Value of the gradient at x0:
 [[-0.03125]
 [ 0.125  ]]
Value of x0:
 [[ 0.03125 ]
 [-0.046875]]
Value of x1:
 [[ 0.0234375]
 [-0.015625 ]]
Value of function f(x0): [[-0.00085449]]
Value of the gradient at x0:
 [[-0.0625  ]
 [ 0.015625]]
Value of x0:
 [[ 0.0234375]
 [-0.015625 ]]
Value of x1:
 [[ 0.0078125 ]
 [-0.01171875]]
Value of function f(x0): [[-0.00021362]]
Value of the gradient at x0:
 [[-0.0078125]
 [ 0.03125  ]]
Value of x0:
 [[ 0.0078125 ]
 [-0.01171875]]
Value of x1:
 [[ 0.00585938]
 [-0.00390625]]
Value of function f(x0): [[ -5.34057617e-05]]
Value of the gradient at x0:
 [[-0.015625  ]
 [ 0.00390625]]
Value of x0:
 [[ 0.00585938]
 [-0.00390625]]
Value of x1:
 [[ 0.00195312]
 [-0.00292969]]
Value of function f(x0): [[ -1.33514404e-05]]
Value of the gradient at x0:
 [[-0.00195312]
 [ 0.0078125 ]]
Value of x0:
 [[ 0.00195312]
 [-0.00292969]]
Value of x1:
 [[ 0.00146484]
 [-0.00097656]]
Value of function f(x0): [[ -3.33786011e-06]]
Value of the gradient at x0:
 [[-0.00390625]
 [ 0.00097656]]
Value of x0:
 [[ 0.00146484]
 [-0.00097656]]
Value of x1:
 [[ 0.00048828]
 [-0.00073242]]


Total number of iterations: 12
Solution value:
 [[ -8.34465027e-07]]
In [54]:
## create 30x30 grid matrix
n = 30
ex1 = np.linspace(-5, 4, num=n)
ex2 = np.linspace(-5, 4, num=n)
grid1, grid2 = np.meshgrid(ex1, ex2)
## fill the grid via looping.  You may prefer to use ufuncs instead.
z = np.empty_like(grid1)
for i in range(z.shape[0]):
    for j in range(z.shape[1]):
        x = np.array([grid1[i,j], grid2[i,j]])
        z[i,j] = f(x,C)[0][0]
        
## Make the plot
plt.figure(figsize=(9,9))
p = plt.contour(grid1, grid2, z,
                levels = [-40, -30,-20, -15, -10, -5, -2, -1, 0, 1, 2, 5, 10, 20, 40],
                colors = 'black')
plt.xlim(-4,8)
plt.ylim(-4,8)
plt.clabel(p, inline=1, fontsize=10)


x0 = np.matrix("2; -3")
R = 0.25
m0, l1 = gradient_ascent(x0,C,R)

## Add a few points
plt.scatter(m0, l1, c='red', s=30)
plt.plot(m0, l1, c='red')
Out[54]:
[<matplotlib.lines.Line2D at 0x1a4f5f6f588>]
In [55]:
alpha = 100
x0 = np.matrix("2; -3")
C = np.matrix(A + np.eye(2) * alpha)
C
Out[55]:
matrix([[ 101.,    1.],
        [   1.,  101.]])
In [56]:
val, vec, condn = eig_cond(C)
print("eigen values:",val)
print("eigen vectors:\n",vec)
print("condition number:",condn)
eigen values: [ 102.  100.]
eigen vectors:
 [[ 0.70710678 -0.70710678]
 [ 0.70710678  0.70710678]]
condition number: 1.00995049384
In [58]:
converge_2d(x0,C,0.005) # Hyperparameter R = 0.005
Value of function f(x0): [[-1301.]]
Value of the gradient at x0:
 [[-398.]
 [ 602.]]
Value of x0:
 [[ 2]
 [-3]]
Value of x1:
 [[ 0.01]
 [ 0.01]]
Value of function f(x0): [[-0.0204]]
Value of the gradient at x0:
 [[-2.04]
 [-2.04]]
Value of x0:
 [[ 0.01]
 [ 0.01]]
Value of x1:
 [[-0.0002]
 [-0.0002]]
Value of function f(x0): [[ -8.16000000e-06]]
Value of the gradient at x0:
 [[ 0.0408]
 [ 0.0408]]
Value of x0:
 [[-0.0002]
 [-0.0002]]
Value of x1:
 [[  4.00000000e-06]
 [  4.00000000e-06]]


Total number of iterations: 3
Solution value:
 [[ -3.26400000e-09]]
In [59]:
## create 30x30 grid matrix
n = 30
ex1 = np.linspace(-6, 4, num=n)
ex2 = np.linspace(-6, 4, num=n)
grid1, grid2 = np.meshgrid(ex1, ex2)
## fill the grid via looping.  You may prefer to use ufuncs instead.
z = np.empty_like(grid1)
for i in range(z.shape[0]):
    for j in range(z.shape[1]):
        x = np.array([grid1[i,j], grid2[i,j]])
        z[i,j] = f(x,C)[0][0]
        
## Make the plot
plt.figure(figsize=(9,9))
p = plt.contour(grid1, grid2, z,
                levels = [-1400,-1300,-1200, -1000,-800, -600, -400, -300, -200, -150, -120, -80, -30, -10, 0],
                colors = 'black')
plt.xlim(-4,4)
plt.ylim(-4,4)
plt.clabel(p, inline=1, fontsize=10)

x0 = np.matrix("2; -3")
R = 0.005
m0, l1 = gradient_ascent(x0,C,R)

## Add a few points
plt.scatter(m0, l1, c='red', s=30)
plt.plot(m0, l1, c='red')
Out[59]:
[<matplotlib.lines.Line2D at 0x1a4f5de3860>]
In [60]:
alpha = 0.01
C = np.matrix(A + np.eye(2) * alpha)
C
Out[60]:
matrix([[ 1.01,  1.  ],
        [ 1.  ,  1.01]])
In [61]:
val, vec, condn = eig_cond(C)
print("eigen values:",val)
print("eigen vectors:\n",vec)
print("condition number:",condn)
eigen values: [ 2.01  0.01]
eigen vectors:
 [[ 0.70710678 -0.70710678]
 [ 0.70710678  0.70710678]]
condition number: 14.1774468788
In [63]:
converge_2d(x0,C,0.45) # Hyperparameter R = 0.45
Value of function f(x0): [[-1.13]]
Value of the gradient at x0:
 [[ 1.96]
 [ 2.06]]
Value of x0:
 [[ 2]
 [-3]]
Value of x1:
 [[ 2.882]
 [-2.073]]
Value of function f(x0): [[-0.78051353]]
Value of the gradient at x0:
 [[-1.67564]
 [-1.57654]]
Value of x0:
 [[ 2.882]
 [-2.073]]
Value of x1:
 [[ 2.127962]
 [-2.782443]]
Value of function f(x0): [[-0.55104749]]
Value of the gradient at x0:
 [[ 1.26640276]
 [ 1.36461086]]
Value of x0:
 [[ 2.127962]
 [-2.782443]]
Value of x1:
 [[ 2.69784324]
 [-2.16836811]]
Value of function f(x0): [[-0.4001457]]
Value of the gradient at x0:
 [[-1.11290712]
 [-1.0155829 ]]
Value of x0:
 [[ 2.69784324]
 [-2.16836811]]
Value of x1:
 [[ 2.19703504]
 [-2.62538042]]
Value of function f(x0): [[-0.30067562]]
Value of the gradient at x0:
 [[ 0.81275006]
 [ 0.90919837]]
Value of x0:
 [[ 2.19703504]
 [-2.62538042]]
Value of x1:
 [[ 2.56277256]
 [-2.21624115]]
Value of function f(x0): [[-0.2348793]]
Value of the gradient at x0:
 [[-0.74431828]
 [-0.648738  ]]
Value of x0:
 [[ 2.56277256]
 [-2.21624115]]
Value of x1:
 [[ 2.22782934]
 [-2.50817325]]
Value of function f(x0): [[-0.19113428]]
Value of the gradient at x0:
 [[ 0.51613124]
 [ 0.61085129]]
Value of x0:
 [[ 2.22782934]
 [-2.50817325]]
Value of x1:
 [[ 2.4600884 ]
 [-2.23329017]]
Value of function f(x0): [[-0.16183363]]
Value of the gradient at x0:
 [[-0.50279822]
 [-0.40893065]]
Value of x0:
 [[ 2.4600884 ]
 [-2.23329017]]
Value of x1:
 [[ 2.2338292 ]
 [-2.41730896]]
Value of function f(x0): [[-0.14199858]]
Value of the gradient at x0:
 [[ 0.32228294]
 [ 0.41530571]]
Value of x0:
 [[ 2.2338292 ]
 [-2.41730896]]
Value of x1:
 [[ 2.37885652]
 [-2.23042139]]
Value of function f(x0): [[-0.12837037]]
Value of the gradient at x0:
 [[-0.34444739]
 [-0.25226183]]
Value of x0:
 [[ 2.37885652]
 [-2.23042139]]
Value of x1:
 [[ 2.2238552 ]
 [-2.34393922]]
Value of function f(x0): [[-0.118816]]
Value of the gradient at x0:
 [[ 0.19569093]
 [ 0.28704682]]
Value of x0:
 [[ 2.2238552 ]
 [-2.34393922]]
Value of x1:
 [[ 2.31191612]
 [-2.21476815]]
Value of function f(x0): [[-0.11193927]]
Value of the gradient at x0:
 [[-0.24053427]
 [-0.15000058]]
Value of x0:
 [[ 2.31191612]
 [-2.21476815]]
Value of x1:
 [[ 2.2036757 ]
 [-2.28226841]]
Value of function f(x0): [[-0.10682617]]
Value of the gradient at x0:
 [[ 0.1131119 ]
 [ 0.20283079]]
Value of x0:
 [[ 2.2036757 ]
 [-2.28226841]]
Value of x1:
 [[ 2.25457606]
 [-2.19099455]]
Value of function f(x0): [[-0.10287831]]
Value of the gradient at x0:
 [[-0.17225452]
 [-0.08334311]]
Value of x0:
 [[ 2.25457606]
 [-2.19099455]]
Value of x1:
 [[ 2.17706152]
 [-2.22849895]]
Value of function f(x0): [[-0.09970385]]
Value of the gradient at x0:
 [[ 0.05933364]
 [ 0.14744485]]
Value of x0:
 [[ 2.17706152]
 [-2.22849895]]
Value of x1:
 [[ 2.20376166]
 [-2.16214877]]
Value of function f(x0): [[-0.09704616]]
Value of the gradient at x0:
 [[-0.127301  ]
 [-0.03998279]]
Value of x0:
 [[ 2.20376166]
 [-2.16214877]]
Value of x1:
 [[ 2.14647621]
 [-2.18014103]]
Value of function f(x0): [[-0.09473707]]
Value of the gradient at x0:
 [[ 0.02440012]
 [ 0.11093247]]
Value of x0:
 [[ 2.14647621]
 [-2.18014103]]
Value of x1:
 [[ 2.15745626]
 [-2.13022142]]
Value of function f(x0): [[-0.09266634]]
Value of the gradient at x0:
 [[-0.09761881]
 [-0.01186526]]
Value of x0:
 [[ 2.15745626]
 [-2.13022142]]
Value of x1:
 [[ 2.1135278 ]
 [-2.13556078]]
Value of function f(x0): [[-0.09076165]]
Value of the gradient at x0:
 [[ 0.00179542]
 [ 0.08677719]]
Value of x0:
 [[ 2.1135278 ]
 [-2.13556078]]
Value of x1:
 [[ 2.11433574]
 [-2.09651105]]
Value of function f(x0): [[-0.08897546]]
Value of the gradient at x0:
 [[-0.07793609]
 [ 0.00628085]]
Value of x0:
 [[ 2.11433574]
 [-2.09651105]]
Value of x1:
 [[ 2.0792645 ]
 [-2.09368467]]
Value of function f(x0): [[-0.0872765]]
Value of the gradient at x0:
 [[-0.01274495]
 [ 0.07071404]]
Value of x0:
 [[ 2.0792645 ]
 [-2.09368467]]
Value of x1:
 [[ 2.07352927]
 [-2.06186335]]
Value of function f(x0): [[-0.08564413]]
Value of the gradient at x0:
 [[-0.06480242]
 [ 0.01790543]]
Value of x0:
 [[ 2.07352927]
 [-2.06186335]]
Value of x1:
 [[ 2.04436818]
 [-2.05380591]]
Value of function f(x0): [[-0.08406467]]
Value of the gradient at x0:
 [[-0.02201191]
 [ 0.05995157]]
Value of x0:
 [[ 2.04436818]
 [-2.05380591]]
Value of x1:
 [[ 2.03446282]
 [-2.0268277 ]]
Value of function f(x0): [[-0.08252899]]
Value of the gradient at x0:
 [[-0.0559595 ]
 [ 0.02526631]]
Value of x0:
 [[ 2.03446282]
 [-2.0268277 ]]
Value of x1:
 [[ 2.00928105]
 [-2.01545786]]
Value of function f(x0): [[-0.08103096]]
Value of the gradient at x0:
 [[-0.02783199]
 [ 0.05266279]]
Value of x0:
 [[ 2.00928105]
 [-2.01545786]]
Value of x1:
 [[ 1.99675665]
 [-1.99175961]]
Value of function f(x0): [[-0.0795664]]
Value of the gradient at x0:
 [[-0.04992922]
 [ 0.02984111]]
Value of x0:
 [[ 1.99675665]
 [-1.99175961]]
Value of x1:
 [[ 1.9742885 ]
 [-1.97833111]]
Value of function f(x0): [[-0.07813243]]
Value of the gradient at x0:
 [[-0.03140056]
 [ 0.04765184]]
Value of x0:
 [[ 1.9742885 ]
 [-1.97833111]]
Value of x1:
 [[ 1.96015825]
 [-1.95688778]]
Value of function f(x0): [[-0.076727]]
Value of the gradient at x0:
 [[-0.0457441 ]
 [ 0.03259682]]
Value of x0:
 [[ 1.96015825]
 [-1.95688778]]
Value of x1:
 [[ 1.9395734 ]
 [-1.94221921]]
Value of function f(x0): [[-0.0753486]]
Value of the gradient at x0:
 [[-0.03349985]
 [ 0.044136  ]]
Value of x0:
 [[ 1.9395734 ]
 [-1.94221921]]
Value of x1:
 [[ 1.92449847]
 [-1.92235801]]
Value of function f(x0): [[-0.07399613]]
Value of the gradient at x0:
 [[-0.04277089]
 [ 0.03416624]]
Value of x0:
 [[ 1.92449847]
 [-1.92235801]]
Value of x1:
 [[ 1.90525157]
 [-1.9069832 ]]
Value of function f(x0): [[-0.07266868]]
Value of the gradient at x0:
 [[-0.03464177]
 [ 0.04160293]]
Value of x0:
 [[ 1.90525157]
 [-1.9069832 ]]
Value of x1:
 [[ 1.88966278]
 [-1.88826189]]
Value of function f(x0): [[-0.07136555]]
Value of the gradient at x0:
 [[-0.04059504]
 [ 0.03496346]]
Value of x0:
 [[ 1.88966278]
 [-1.88826189]]
Value of x1:
 [[ 1.87139501]
 [-1.87252833]]
Value of function f(x0): [[-0.0700861]]
Value of the gradient at x0:
 [[-0.03516126]
 [ 0.03971721]]
Value of x0:
 [[ 1.87139501]
 [-1.87252833]]
Value of x1:
 [[ 1.85557244]
 [-1.85465559]]
Value of function f(x0): [[-0.06882981]]
Value of the gradient at x0:
 [[-0.03894516]
 [ 0.0352594 ]]
Value of x0:
 [[ 1.85557244]
 [-1.85465559]]
Value of x1:
 [[ 1.83804712]
 [-1.83878886]]
Value of function f(x0): [[-0.06759617]]
Value of the gradient at x0:
 [[-0.03527747]
 [ 0.03825925]]
Value of x0:
 [[ 1.83804712]
 [-1.83878886]]
Value of x1:
 [[ 1.82217226]
 [-1.8215722 ]]
Value of function f(x0): [[-0.06638473]]
Value of the gradient at x0:
 [[-0.03764358]
 [ 0.03523131]]
Value of x0:
 [[ 1.82217226]
 [-1.8215722 ]]
Value of x1:
 [[ 1.80523265]
 [-1.8057181 ]]
Value of function f(x0): [[-0.06519506]]
Value of the gradient at x0:
 [[-0.03513375]
 [ 0.03708527]]
Value of x0:
 [[ 1.80523265]
 [-1.8057181 ]]
Value of x1:
 [[ 1.78942247]
 [-1.78902973]]
Value of function f(x0): [[-0.06402676]]
Value of the gradient at x0:
 [[-0.03657391]
 [ 0.03499513]]
Value of x0:
 [[ 1.78942247]
 [-1.78902973]]
Value of x1:
 [[ 1.7729642 ]
 [-1.77328192]]
Value of function f(x0): [[-0.06287941]]
Value of the gradient at x0:
 [[-0.03482385]
 [ 0.03610108]]
Value of x0:
 [[ 1.7729642 ]
 [-1.77328192]]
Value of x1:
 [[ 1.75729347]
 [-1.75703644]]
Value of function f(x0): [[-0.06175264]]
Value of the gradient at x0:
 [[-0.03565994]
 [ 0.03462666]]
Value of x0:
 [[ 1.75729347]
 [-1.75703644]]
Value of x1:
 [[ 1.7412465 ]
 [-1.74145444]]
Value of function f(x0): [[-0.06064607]]
Value of the gradient at x0:
 [[-0.03440905]
 [ 0.03524497]]
Value of x0:
 [[ 1.7412465 ]
 [-1.74145444]]
Value of x1:
 [[ 1.72576243]
 [-1.72559421]]
Value of function f(x0): [[-0.05955934]]
Value of the gradient at x0:
 [[-0.0348517 ]
 [ 0.03417544]]
Value of x0:
 [[ 1.72576243]
 [-1.72559421]]
Value of x1:
 [[ 1.71007917]
 [-1.71021526]]
Value of function f(x0): [[-0.05849209]]
Value of the gradient at x0:
 [[-0.0339294 ]
 [ 0.03447649]]
Value of x0:
 [[ 1.71007917]
 [-1.71021526]]
Value of x1:
 [[ 1.69481094]
 [-1.69470084]]
Value of function f(x0): [[-0.05744396]]
Value of the gradient at x0:
 [[-0.03411642]
 [ 0.03367382]]
Value of x0:
 [[ 1.69481094]
 [-1.69470084]]
Value of x1:
 [[ 1.67945855]
 [-1.67954762]]
Value of function f(x0): [[-0.05641462]]
Value of the gradient at x0:
 [[-0.03341103]
 [ 0.03376909]]
Value of x0:
 [[ 1.67945855]
 [-1.67954762]]
Value of x1:
 [[ 1.66442359]
 [-1.66435153]]
Value of function f(x0): [[-0.05540372]]
Value of the gradient at x0:
 [[-0.03343259]
 [ 0.03314291]]
Value of x0:
 [[ 1.66442359]
 [-1.66435153]]
Value of x1:
 [[ 1.64937892]
 [-1.64943722]]
Value of function f(x0): [[-0.05441094]]
Value of the gradient at x0:
 [[-0.03287099]
 [ 0.03310533]]
Value of x0:
 [[ 1.64937892]
 [-1.64943722]]
Value of x1:
 [[ 1.63458698]
 [-1.63453982]]
Value of function f(x0): [[-0.05343595]]
Value of the gradient at x0:
 [[-0.03278606]
 [ 0.03259647]]
Value of x0:
 [[ 1.63458698]
 [-1.63453982]]
Value of x1:
 [[ 1.61983325]
 [-1.6198714 ]]
Value of function f(x0): [[-0.05247843]]
Value of the gradient at x0:
 [[-0.03232036]
 [ 0.03247373]]
Value of x0:
 [[ 1.61983325]
 [-1.6198714 ]]
Value of x1:
 [[ 1.60528909]
 [-1.60525822]]
Value of function f(x0): [[-0.05153807]]
Value of the gradient at x0:
 [[-0.03216751]
 [ 0.03204343]]
Value of x0:
 [[ 1.60528909]
 [-1.60525822]]
Value of x1:
 [[ 1.59081371]
 [-1.59083868]]
Value of function f(x0): [[-0.05061456]]
Value of the gradient at x0:
 [[-0.03176633]
 [ 0.03186671]]
Value of x0:
 [[ 1.59081371]
 [-1.59083868]]
Value of x1:
 [[ 1.57651886]
 [-1.57649866]]
Value of function f(x0): [[-0.0497076]]
Value of the gradient at x0:
 [[-0.03157078]
 [ 0.03148957]]
Value of x0:
 [[ 1.57651886]
 [-1.57649866]]
Value of x1:
 [[ 1.56231201]
 [-1.56232835]]
Value of function f(x0): [[-0.04881689]]
Value of the gradient at x0:
 [[-0.03121355]
 [ 0.03127925]]
Value of x0:
 [[ 1.56231201]
 [-1.56232835]]
Value of x1:
 [[ 1.54826591]
 [-1.54825269]]
Value of function f(x0): [[-0.04794214]]
Value of the gradient at x0:
 [[-0.03099176]
 [ 0.03093861]]
Value of x0:
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Value of function f(x0): [[-0.00012064]]
Value of the gradient at x0:
 [[-0.00155329]
 [ 0.00155329]]
Value of x0:
 [[ 0.07766464]
 [-0.07766464]]
Value of x1:
 [[ 0.07696566]
 [-0.07696566]]
Value of function f(x0): [[-0.00011847]]
Value of the gradient at x0:
 [[-0.00153931]
 [ 0.00153931]]
Value of x0:
 [[ 0.07696566]
 [-0.07696566]]
Value of x1:
 [[ 0.07627297]
 [-0.07627297]]
Value of function f(x0): [[-0.00011635]]
Value of the gradient at x0:
 [[-0.00152546]
 [ 0.00152546]]
Value of x0:
 [[ 0.07627297]
 [-0.07627297]]
Value of x1:
 [[ 0.07558651]
 [-0.07558651]]
Value of function f(x0): [[-0.00011427]]
Value of the gradient at x0:
 [[-0.00151173]
 [ 0.00151173]]
Value of x0:
 [[ 0.07558651]
 [-0.07558651]]
Value of x1:
 [[ 0.07490623]
 [-0.07490623]]
Value of function f(x0): [[-0.00011222]]
Value of the gradient at x0:
 [[-0.00149812]
 [ 0.00149812]]
Value of x0:
 [[ 0.07490623]
 [-0.07490623]]
Value of x1:
 [[ 0.07423208]
 [-0.07423208]]
Value of function f(x0): [[-0.00011021]]
Value of the gradient at x0:
 [[-0.00148464]
 [ 0.00148464]]
Value of x0:
 [[ 0.07423208]
 [-0.07423208]]
Value of x1:
 [[ 0.07356399]
 [-0.07356399]]
Value of function f(x0): [[-0.00010823]]
Value of the gradient at x0:
 [[-0.00147128]
 [ 0.00147128]]
Value of x0:
 [[ 0.07356399]
 [-0.07356399]]
Value of x1:
 [[ 0.07290191]
 [-0.07290191]]
Value of function f(x0): [[-0.00010629]]
Value of the gradient at x0:
 [[-0.00145804]
 [ 0.00145804]]
Value of x0:
 [[ 0.07290191]
 [-0.07290191]]
Value of x1:
 [[ 0.07224579]
 [-0.07224579]]
Value of function f(x0): [[-0.00010439]]
Value of the gradient at x0:
 [[-0.00144492]
 [ 0.00144492]]
Value of x0:
 [[ 0.07224579]
 [-0.07224579]]
Value of x1:
 [[ 0.07159558]
 [-0.07159558]]
Value of function f(x0): [[-0.00010252]]
Value of the gradient at x0:
 [[-0.00143191]
 [ 0.00143191]]
Value of x0:
 [[ 0.07159558]
 [-0.07159558]]
Value of x1:
 [[ 0.07095122]
 [-0.07095122]]
Value of function f(x0): [[-0.00010068]]
Value of the gradient at x0:
 [[-0.00141902]
 [ 0.00141902]]
Value of x0:
 [[ 0.07095122]
 [-0.07095122]]
Value of x1:
 [[ 0.07031266]
 [-0.07031266]]
Value of function f(x0): [[ -9.88774042e-05]]
Value of the gradient at x0:
 [[-0.00140625]
 [ 0.00140625]]
Value of x0:
 [[ 0.07031266]
 [-0.07031266]]
Value of x1:
 [[ 0.06967985]
 [-0.06967985]]
Value of function f(x0): [[ -9.71056200e-05]]
Value of the gradient at x0:
 [[-0.0013936]
 [ 0.0013936]]
Value of x0:
 [[ 0.06967985]
 [-0.06967985]]
Value of x1:
 [[ 0.06905273]
 [-0.06905273]]
Value of function f(x0): [[ -9.53655844e-05]]
Value of the gradient at x0:
 [[-0.00138105]
 [ 0.00138105]]
Value of x0:
 [[ 0.06905273]
 [-0.06905273]]
Value of x1:
 [[ 0.06843125]
 [-0.06843125]]
Value of function f(x0): [[ -9.36567285e-05]]
Value of the gradient at x0:
 [[-0.00136863]
 [ 0.00136863]]
Value of x0:
 [[ 0.06843125]
 [-0.06843125]]
Value of x1:
 [[ 0.06781537]
 [-0.06781537]]
Value of function f(x0): [[ -9.19784935e-05]]
Value of the gradient at x0:
 [[-0.00135631]
 [ 0.00135631]]
Value of x0:
 [[ 0.06781537]
 [-0.06781537]]
Value of x1:
 [[ 0.06720503]
 [-0.06720503]]
Value of function f(x0): [[ -9.03303309e-05]]
Value of the gradient at x0:
 [[-0.0013441]
 [ 0.0013441]]
Value of x0:
 [[ 0.06720503]
 [-0.06720503]]
Value of x1:
 [[ 0.06660019]
 [-0.06660019]]
Value of function f(x0): [[ -8.87117017e-05]]
Value of the gradient at x0:
 [[-0.001332]
 [ 0.001332]]
Value of x0:
 [[ 0.06660019]
 [-0.06660019]]
Value of x1:
 [[ 0.06600079]
 [-0.06600079]]
Value of function f(x0): [[ -8.71220767e-05]]
Value of the gradient at x0:
 [[-0.00132002]
 [ 0.00132002]]
Value of x0:
 [[ 0.06600079]
 [-0.06600079]]
Value of x1:
 [[ 0.06540678]
 [-0.06540678]]
Value of function f(x0): [[ -8.55609362e-05]]
Value of the gradient at x0:
 [[-0.00130814]
 [ 0.00130814]]
Value of x0:
 [[ 0.06540678]
 [-0.06540678]]
Value of x1:
 [[ 0.06481812]
 [-0.06481812]]
Value of function f(x0): [[ -8.40277698e-05]]
Value of the gradient at x0:
 [[-0.00129636]
 [ 0.00129636]]
Value of x0:
 [[ 0.06481812]
 [-0.06481812]]
Value of x1:
 [[ 0.06423476]
 [-0.06423476]]
Value of function f(x0): [[ -8.25220762e-05]]
Value of the gradient at x0:
 [[-0.0012847]
 [ 0.0012847]]
Value of x0:
 [[ 0.06423476]
 [-0.06423476]]
Value of x1:
 [[ 0.06365664]
 [-0.06365664]]
Value of function f(x0): [[ -8.10433631e-05]]
Value of the gradient at x0:
 [[-0.00127313]
 [ 0.00127313]]
Value of x0:
 [[ 0.06365664]
 [-0.06365664]]
Value of x1:
 [[ 0.06308373]
 [-0.06308373]]
Value of function f(x0): [[ -7.95911471e-05]]
Value of the gradient at x0:
 [[-0.00126167]
 [ 0.00126167]]
Value of x0:
 [[ 0.06308373]
 [-0.06308373]]
Value of x1:
 [[ 0.06251598]
 [-0.06251598]]
Value of function f(x0): [[ -7.81649533e-05]]
Value of the gradient at x0:
 [[-0.00125032]
 [ 0.00125032]]
Value of x0:
 [[ 0.06251598]
 [-0.06251598]]
Value of x1:
 [[ 0.06195334]
 [-0.06195334]]
Value of function f(x0): [[ -7.67643155e-05]]
Value of the gradient at x0:
 [[-0.00123907]
 [ 0.00123907]]
Value of x0:
 [[ 0.06195334]
 [-0.06195334]]
Value of x1:
 [[ 0.06139576]
 [-0.06139576]]
Value of function f(x0): [[ -7.53887758e-05]]
Value of the gradient at x0:
 [[-0.00122792]
 [ 0.00122792]]
Value of x0:
 [[ 0.06139576]
 [-0.06139576]]
Value of x1:
 [[ 0.06084319]
 [-0.06084319]]
Value of function f(x0): [[ -7.40378843e-05]]
Value of the gradient at x0:
 [[-0.00121686]
 [ 0.00121686]]
Value of x0:
 [[ 0.06084319]
 [-0.06084319]]
Value of x1:
 [[ 0.0602956]
 [-0.0602956]]
Value of function f(x0): [[ -7.27111995e-05]]
Value of the gradient at x0:
 [[-0.00120591]
 [ 0.00120591]]
Value of x0:
 [[ 0.0602956]
 [-0.0602956]]
Value of x1:
 [[ 0.05975294]
 [-0.05975294]]
Value of function f(x0): [[ -7.14082875e-05]]
Value of the gradient at x0:
 [[-0.00119506]
 [ 0.00119506]]
Value of x0:
 [[ 0.05975294]
 [-0.05975294]]
Value of x1:
 [[ 0.05921517]
 [-0.05921517]]
Value of function f(x0): [[ -7.01287224e-05]]
Value of the gradient at x0:
 [[-0.0011843]
 [ 0.0011843]]
Value of x0:
 [[ 0.05921517]
 [-0.05921517]]
Value of x1:
 [[ 0.05868223]
 [-0.05868223]]
Value of function f(x0): [[ -6.88720858e-05]]
Value of the gradient at x0:
 [[-0.00117364]
 [ 0.00117364]]
Value of x0:
 [[ 0.05868223]
 [-0.05868223]]
Value of x1:
 [[ 0.05815409]
 [-0.05815409]]
Value of function f(x0): [[ -6.76379669e-05]]
Value of the gradient at x0:
 [[-0.00116308]
 [ 0.00116308]]
Value of x0:
 [[ 0.05815409]
 [-0.05815409]]
Value of x1:
 [[ 0.0576307]
 [-0.0576307]]
Value of function f(x0): [[ -6.64259622e-05]]
Value of the gradient at x0:
 [[-0.00115261]
 [ 0.00115261]]
Value of x0:
 [[ 0.0576307]
 [-0.0576307]]
Value of x1:
 [[ 0.05711203]
 [-0.05711203]]
Value of function f(x0): [[ -6.52356753e-05]]
Value of the gradient at x0:
 [[-0.00114224]
 [ 0.00114224]]
Value of x0:
 [[ 0.05711203]
 [-0.05711203]]
Value of x1:
 [[ 0.05659802]
 [-0.05659802]]
Value of function f(x0): [[ -6.40667173e-05]]
Value of the gradient at x0:
 [[-0.00113196]
 [ 0.00113196]]
Value of x0:
 [[ 0.05659802]
 [-0.05659802]]
Value of x1:
 [[ 0.05608864]
 [-0.05608864]]
Value of function f(x0): [[ -6.29187058e-05]]
Value of the gradient at x0:
 [[-0.00112177]
 [ 0.00112177]]
Value of x0:
 [[ 0.05608864]
 [-0.05608864]]
Value of x1:
 [[ 0.05558384]
 [-0.05558384]]
Value of function f(x0): [[ -6.17912655e-05]]
Value of the gradient at x0:
 [[-0.00111168]
 [ 0.00111168]]
Value of x0:
 [[ 0.05558384]
 [-0.05558384]]
Value of x1:
 [[ 0.05508359]
 [-0.05508359]]
Value of function f(x0): [[ -6.06840278e-05]]
Value of the gradient at x0:
 [[-0.00110167]
 [ 0.00110167]]
Value of x0:
 [[ 0.05508359]
 [-0.05508359]]
Value of x1:
 [[ 0.05458783]
 [-0.05458783]]
Value of function f(x0): [[ -5.95966307e-05]]
Value of the gradient at x0:
 [[-0.00109176]
 [ 0.00109176]]
Value of x0:
 [[ 0.05458783]
 [-0.05458783]]
Value of x1:
 [[ 0.05409654]
 [-0.05409654]]
Value of function f(x0): [[ -5.85287187e-05]]
Value of the gradient at x0:
 [[-0.00108193]
 [ 0.00108193]]
Value of x0:
 [[ 0.05409654]
 [-0.05409654]]
Value of x1:
 [[ 0.05360967]
 [-0.05360967]]
Value of function f(x0): [[ -5.74799426e-05]]
Value of the gradient at x0:
 [[-0.00107219]
 [ 0.00107219]]
Value of x0:
 [[ 0.05360967]
 [-0.05360967]]
Value of x1:
 [[ 0.05312719]
 [-0.05312719]]
Value of function f(x0): [[ -5.64499595e-05]]
Value of the gradient at x0:
 [[-0.00106254]
 [ 0.00106254]]
Value of x0:
 [[ 0.05312719]
 [-0.05312719]]
Value of x1:
 [[ 0.05264904]
 [-0.05264904]]


Total number of iterations: 427
Solution value:
 [[ -5.54384326e-05]]
In [64]:
## create 30x30 grid matrix
n = 30
ex1 = np.linspace(-6, 3, num=n)
ex2 = np.linspace(-6, 3, num=n)
grid1, grid2 = np.meshgrid(ex1, ex2)
## fill the grid via looping.  You may prefer to use ufuncs instead.
z = np.empty_like(grid1)
for i in range(z.shape[0]):
    for j in range(z.shape[1]):
        x = np.array([grid1[i,j], grid2[i,j]])
        z[i,j] = f(x,C)[0][0]
        
## Make the plot
plt.figure(figsize=(9,9))
p = plt.contour(grid1, grid2, z,
                levels = [-3,-2.5,-2,-1.5,-1, 0, 1],
                colors = 'black')
plt.xlim(-4,4)
plt.ylim(-4,4)
plt.clabel(p, inline=1, fontsize=10)

x0 = np.matrix("2; -3")
R = 0.45
m0, l1 = gradient_ascent(x0,C,R)

## Add a few points
plt.scatter(m0, l1, c='red', s=30)
plt.plot(m0, l1, c='red')
Out[64]:
[<matplotlib.lines.Line2D at 0x1a4f6441860>]

We have following observations for the 3 different alpha values:

  • With alpha = 1, the matrix C is matrix([[ 2., 1.],[ 1., 2.]]) and the condition number is 1.73205080757. The hyperparameter selected is R = 0.25 and the number of iterations for convergence is 12.
  • With alpha = 100, the matrix C is matrix([[ 101., 1.],[ 1., 101.]]) and the condition number is 1.00995049384. The hyperparameter selected is R = 0.005 and the number of iterations for convergence is 3.
  • With alpha = 0.01, the matrix C is matrix([[ 1.01, 1. ],[ 1. , 1.01]]) and the condition number is 14.1774468788. The hyperparameter selected is R = 0.45 and the number of iterations for convergence is 427.

Based on the above results, we can say that higher the condition number, the more time it takes to achieve convergence for a given function.

2.1. Pick your number of repetitions R (1000 is a good choice), and sample sizes N (10, 1000, 100,000) are good choices, but you may want to adjust if the latter is too slow).

In [65]:
R = 1000
N = 10

2.1.2. For R times create sample of N of standard normals, and take the mean of it. You'll have R means of N standard normals.

In [66]:
m = np.random.normal(size=(R,N)).mean(axis=1)
print("Means:",m)
Means: [ -1.68351538e-01  -3.24532131e-01  -3.23605062e-01   3.27915793e-01
  -3.29173347e-01   4.44049280e-01   4.75626867e-03  -2.21817802e-01
  -1.03206850e-01   2.28380776e-03   5.59281052e-01  -3.73196551e-01
   2.15354376e-01   3.17590107e-01  -6.95910861e-02  -1.41989179e-01
   6.40728914e-01   2.55259809e-01   3.80664831e-01  -3.85351090e-01
  -3.85894783e-01   1.54134019e-02   1.32184793e-01  -7.69602587e-02
  -2.48824816e-01  -1.28053111e-01   7.41980272e-01  -6.41612149e-02
  -9.42852850e-03  -4.08938851e-02   3.46034725e-01  -9.96948358e-01
   3.74441060e-01   1.50195157e-01  -1.67868804e-01   6.56149543e-02
  -3.20851438e-01  -1.93891319e-01  -4.84594289e-01  -1.21371117e-01
  -9.12088072e-02   3.86838175e-01   2.51295737e-01  -1.94263399e-01
   1.38998628e-01  -3.46715301e-01  -4.55219917e-01   3.98198259e-01
  -1.84243592e-01  -4.11105688e-02  -1.00355485e-01  -2.00372316e-02
  -8.85428028e-02  -1.10540215e-01   3.76773088e-01   4.55958978e-01
  -5.91515226e-01  -5.04754227e-01  -4.80281246e-02  -4.28970607e-01
  -7.63856959e-01   2.20114008e-01   5.57663574e-01   4.35216908e-01
  -1.82970094e-01  -2.97020817e-01   1.22020355e-01   4.37097258e-01
  -7.06521428e-01   1.24439652e-01   2.26209123e-01   2.53645514e-01
   4.87519342e-01  -2.80621897e-01   1.50758341e-03   2.27181483e-01
   8.58662695e-01  -3.71724300e-01  -5.81446254e-01  -1.21728914e-01
  -5.32286001e-01  -2.15236055e-01  -4.22139707e-01  -2.22431772e-01
  -1.60120532e-01  -3.90212114e-01  -7.08080928e-01  -4.24133509e-01
   6.12222126e-02  -6.58000332e-01  -2.40031505e-01   4.73539113e-01
  -5.14492026e-01   2.44082642e-02   1.50643423e-02  -8.62930424e-02
  -4.12421301e-02   2.44861389e-01   2.36574490e-02  -3.71810394e-01
   4.27451431e-02  -3.12061016e-01  -1.31318224e-02  -2.98785967e-01
  -9.44165089e-02  -1.15341543e-01   8.72240751e-01  -2.00962440e-02
  -2.20665354e-01  -1.82570310e-02   4.77625520e-03  -2.16614785e-02
   2.90851952e-01   2.90171721e-01  -1.19907418e-01  -1.23958914e-02
  -2.19477137e-01   7.53856626e-02   6.31666798e-01  -2.99642941e-01
  -2.63356453e-01   6.95351560e-02   2.10847184e-01   4.68088193e-01
   2.62998453e-01   3.38586145e-01   3.00376496e-01   3.42958709e-01
  -2.24926715e-01  -3.14814222e-01  -3.58256076e-01   4.00545676e-01
  -3.06034908e-01   4.28270238e-01   2.53333515e-01   5.73320096e-01
   5.16769176e-01   5.57826736e-01  -3.55495000e-02  -9.01449769e-02
   2.34376327e-03  -2.74393781e-01  -6.12704672e-01   2.76987470e-02
  -9.60699732e-02  -1.41394453e-01  -8.52526982e-02   7.15551945e-02
  -3.48631784e-01  -2.47052346e-01  -8.63098211e-02   2.41068240e-01
  -1.82554820e-01  -2.22696557e-01  -2.04630443e-01  -2.27016327e-01
  -5.54384060e-01  -1.43583137e-01  -3.73285778e-01  -3.23769650e-01
  -2.64238495e-01   4.72781800e-01   5.15488368e-02  -5.08623021e-01
  -5.88352676e-01  -6.85730812e-02  -1.00646308e-01   5.24921091e-02
   2.11020586e-01  -3.53094256e-01  -7.80581977e-01  -3.67031116e-01
   1.58740580e-03  -3.60334471e-01   9.96139558e-02  -1.74797015e-01
   5.20267155e-02   2.00460208e-02   2.65451737e-01   1.25053376e-01
  -2.89533586e-01  -2.25297322e-01  -2.59398544e-01  -1.64808751e-01
   3.48875651e-01   2.31883024e-01  -5.35487089e-01  -6.70548460e-02
   5.83129228e-01  -1.64611673e-01  -1.87460545e-01  -1.47228743e-01
   3.27167722e-01   5.06976596e-01   2.25343476e-01  -4.11460014e-01
  -1.93095034e-01  -7.57630473e-02  -2.62885050e-01   2.16215307e-02
  -1.00274766e-01  -1.68144204e-01   4.11010857e-01   4.13212746e-01
   3.77008773e-01  -4.15475107e-02  -1.14150600e-01  -3.46725169e-01
  -2.98621350e-02   4.92396840e-01  -4.47804760e-01  -8.18168475e-01
  -4.29918526e-01  -1.16334960e-01  -6.55287245e-03   2.56201671e-01
  -4.63527154e-01  -2.10035907e-01  -4.03282174e-01   9.03552433e-02
  -9.42733150e-02   2.50231255e-01   3.18065423e-01   1.40253139e-01
   5.52099650e-01   1.94482762e-01   6.52091670e-01   4.50273687e-01
   1.58533521e-01  -1.79558791e-01   4.50516695e-01  -2.17726922e-01
  -9.53623719e-02  -3.31592720e-01  -5.36084281e-01   6.70485574e-01
   1.57564935e-01  -2.98407330e-01   1.38844472e-01   4.81011678e-01
   4.98224499e-02  -3.25336026e-01  -5.39369912e-01   8.06726272e-03
   3.28470144e-01  -5.16944411e-01   1.39759310e-01   6.13768092e-01
   1.76235313e-01  -1.10596345e+00   2.67375492e-02   4.53822711e-01
  -3.45398518e-01  -1.79443226e-01   5.41952427e-01  -3.49126347e-01
   3.33553208e-01  -5.74319519e-01   5.62585268e-01   2.26368397e-01
  -2.92677079e-01   3.25975928e-01   2.09010758e-01  -3.40175980e-01
   1.53280575e-01   3.51839420e-02  -3.53886551e-01   4.10181814e-01
   4.13248961e-01   1.74216917e-01  -3.89535996e-01  -1.12248017e-01
   1.33443522e-01   1.06686466e-02   1.72392939e-01   3.03806342e-02
   6.46706348e-02  -2.94627144e-01   4.29598977e-01  -7.10670097e-02
  -8.35269071e-02  -2.53132677e-02   7.80137788e-02  -4.54042001e-01
  -3.81666946e-02  -4.51188914e-01   2.48618448e-01   2.61138606e-01
   1.34706723e-01  -2.56224143e-01  -3.86570089e-01   5.07645786e-01
   1.37204277e-01   8.21419884e-01  -2.43334570e-01   2.75264366e-01
   3.77970202e-01   8.18412293e-01   3.33781924e-01  -1.25248147e-01
   4.44299542e-01   1.77309813e-01   3.15795336e-02   1.63950262e-01
  -4.30315122e-01   3.63037191e-02   1.30195284e-01  -2.38333871e-01
   1.83301424e-01   2.62078118e-01   5.04296735e-02  -1.18168469e-01
  -1.80833992e-02   1.06967744e-01  -1.29590384e-01   6.17930778e-05
   3.86447670e-01   3.55759155e-01   3.53953372e-01  -4.16751094e-02
  -6.51978691e-02  -4.41865254e-01   5.02265674e-01  -7.18942432e-02
  -1.54000386e-01   1.29353881e-01   1.56334646e-01  -6.69296504e-02
   3.27661915e-01  -2.79730813e-01  -5.78943011e-03   4.16009522e-02
  -2.24270784e-01  -7.75530851e-02  -1.26256698e-01   1.18121580e-01
  -4.33515783e-02  -3.26674347e-02   5.71349599e-02  -3.85084336e-01
   5.21891210e-01  -1.34696994e-01   7.33852216e-01  -3.76386218e-01
   2.40016312e-01   3.67205097e-01   1.87990517e-01   2.77378507e-03
  -4.63378096e-01   7.69848368e-02   1.05183436e-01  -2.47602500e-02
  -6.16085672e-01  -4.66975226e-01   9.28546329e-03   2.03079558e-01
   2.18701589e-01   5.99645234e-02   3.86831436e-01  -6.00573236e-02
  -5.86499041e-01  -1.36178180e-01  -2.56973543e-01   1.22677190e-01
  -5.27387558e-01  -3.02394501e-01   1.10900539e-01  -6.90824935e-02
  -1.90422591e-01  -5.58586911e-02   1.95287554e-01   2.39710442e-02
  -3.76367255e-01  -6.05577616e-01   1.81822587e-01  -3.26517587e-01
  -4.01319773e-01   2.40376968e-01   5.01466491e-01   1.62267968e-01
  -1.70757747e-02   5.45465195e-03   1.45212893e-01   4.54968110e-01
   1.36044268e-01  -3.38927335e-01  -1.16343082e-01   2.88429175e-01
  -1.40118723e-01   3.60993497e-02  -4.47269010e-01   8.48755580e-02
   3.36972409e-01  -6.19117587e-02   7.93817745e-02  -4.95740027e-01
   4.08238595e-03   7.17898807e-02   1.63993003e-03   2.69679149e-01
  -3.67384413e-01  -2.22937051e-01  -5.47712755e-03  -3.30721325e-01
  -3.15949767e-02   1.53486379e-01   1.67860469e-02   9.22510769e-02
   2.20367433e-01   2.80201852e-01  -4.27064669e-01  -1.67936464e-01
   1.49153511e-02  -2.28242589e-01  -4.05804920e-01   1.66931754e-01
  -1.83072465e-01  -5.13018134e-01   1.59828163e-01  -1.77823415e-03
   2.55519396e-01  -1.10292990e-01  -5.46968767e-01   3.27865759e-01
   2.95563384e-01  -3.38073647e-01  -3.45265085e-02  -2.03624641e-01
  -5.67916777e-02  -1.80880144e-01  -6.53505780e-02   1.69134469e-01
   3.24924909e-01   2.28148816e-01  -2.28965859e-02  -4.98846783e-01
   2.45377340e-01   4.10207464e-01   1.40329673e-01  -3.72401065e-01
  -4.23401369e-01  -2.17329274e-01   1.92099860e-01  -6.17643171e-02
  -3.28572665e-01   1.03307821e-01  -6.28426912e-02  -1.18724797e-01
  -7.31008929e-02   4.11249269e-01   8.68970022e-03  -4.05044423e-01
   2.90530896e-01  -4.35333900e-02   1.68863961e-01   5.54427907e-02
  -4.63063778e-01  -5.35793163e-01   2.39983849e-01  -4.45242054e-01
  -3.14623361e-02   3.76022924e-01   2.49495983e-01  -1.39280820e-02
   3.13618522e-01  -1.38427274e-01  -2.01533399e-01   8.20748285e-02
   3.62058954e-02   4.35875443e-01  -1.63941065e-01   2.14820613e-01
  -1.98281876e-01  -4.82850055e-01   2.95660001e-01  -4.80381754e-01
   3.23792007e-01   1.02630324e-01  -4.54866038e-01   1.38637478e-01
  -2.92405100e-01   4.03003094e-01  -1.95261523e-02   5.05517848e-01
  -1.26403395e-02   4.16426306e-01   3.71810794e-01   2.39936180e-01
  -1.43146301e-01   2.84174498e-01   4.78651707e-01   2.68879773e-01
  -2.89511954e-02  -1.17339015e-01  -1.18710927e-01  -3.40140050e-01
   1.32352646e-02  -2.64749743e-01  -5.52505745e-01   2.13856469e-01
  -7.16517677e-01   2.46796704e-01   1.86294898e-01   8.59842498e-02
   2.13847581e-01   6.16517081e-02   5.88496934e-01   3.71051942e-01
   3.66482094e-01   1.39436260e-01  -1.71585196e-01   1.82125047e-01
   3.98634840e-02  -3.31369298e-02   1.36107653e-01   3.24182423e-01
  -4.91561634e-01  -4.98215967e-01  -8.38530701e-02  -3.91093290e-01
   2.25783062e-02  -1.49149176e-01  -1.40234803e-01   2.87932417e-01
  -3.53747256e-01  -6.36057128e-02   1.26917210e-01   1.47846828e-01
   4.42338347e-01  -4.07841756e-01   2.94216633e-01  -2.45816294e-01
  -2.87607988e-01   1.22156045e-01  -1.12373184e-01   3.81909802e-01
  -1.07460190e-01   3.74311801e-01   3.58336434e-01   4.15305074e-03
  -3.90399276e-01   5.13761476e-01  -3.48800647e-01   4.15780694e-01
   2.71511990e-01  -2.20899397e-01   4.84738484e-02   2.25892236e-01
  -9.66526180e-02   5.37338492e-03   2.67344354e-01  -2.04174836e-01
  -5.93653678e-02   7.44301395e-02   6.78188157e-02  -2.34135849e-01
  -1.90508330e-01   4.59126621e-01   5.32032452e-01  -1.76892992e-01
   1.08491334e-01   4.85157551e-02  -1.22236499e-01  -3.71578846e-01
   9.26617510e-02  -1.24914557e-01  -1.38978041e-03   2.20496783e-01
  -2.73252596e-01   3.29904604e-01   8.70995660e-02  -7.05999329e-02
  -9.94031449e-02   2.27662580e-01   1.75595415e-01   2.42296475e-01
   4.20202218e-01   6.23072520e-02   1.08354661e-01  -3.47533831e-02
  -3.58637450e-01  -3.74249339e-01   6.88239499e-02  -9.07489865e-02
   3.26811353e-01   3.39546995e-01   3.16920568e-01  -3.77359261e-02
   2.12142082e-02  -1.49381140e-01  -4.31561140e-02  -3.55358997e-01
  -1.52333280e-01  -7.75878910e-02  -6.20555405e-02  -8.34963839e-02
   3.17760600e-01   2.61694598e-02  -6.19546096e-01  -1.02512006e-01
  -4.06031334e-02  -1.23469171e-01   3.91939650e-01   5.64905708e-01
   8.26279885e-02  -4.53679480e-01  -5.67344597e-01   1.58889863e-01
  -3.27424159e-02  -5.95936780e-02  -6.61942315e-02  -1.91079328e-01
   1.86233551e-01  -9.18532361e-02  -7.21276643e-02  -9.18721578e-02
   3.37399197e-01  -3.97397969e-01   6.84308763e-02   7.00309775e-02
   2.29765798e-01   3.01969849e-01  -8.21132741e-01  -2.64946507e-01
  -6.72448884e-01  -6.22213038e-01   1.00833076e-01   9.03247978e-02
   1.43026359e-01   5.56102012e-03  -5.96456868e-01   9.25917369e-02
   5.03701141e-01   4.86436240e-01   4.89182751e-02   3.27333298e-01
   4.74306385e-01   3.50643983e-01  -3.98481925e-01   2.54995633e-01
  -1.43577847e-01   5.89163064e-02   5.02766540e-01   3.20809239e-01
   3.38684364e-01   1.08955029e-01  -1.54016048e-01  -2.66772304e-01
  -2.90008698e-02   4.00038362e-01  -3.30095487e-01  -4.55795314e-02
   1.06517285e-01   5.40686555e-01   5.61414513e-02  -3.42682757e-01
   4.14379016e-01   2.32195854e-01  -1.35144380e-01   5.75893810e-01
   8.60506052e-02  -3.09826456e-01   4.62428874e-01   2.84705805e-01
  -1.70058208e-02   2.83270468e-01  -1.07825262e-01  -4.62339312e-01
  -1.36554878e-01   1.35038290e-01  -6.13285337e-02   8.07839814e-02
   2.16530824e-01  -2.20188148e-01  -1.03495786e-01  -2.25813228e-01
   1.22934858e-01   4.69154551e-02  -3.12426549e-02   2.65871600e-01
  -1.02429944e-01   7.93136444e-01   4.43974743e-02  -9.21908752e-02
   1.92074599e-01  -3.79107421e-01  -9.64452521e-02   2.37691413e-01
   6.86198911e-02   3.79780399e-01   5.98141159e-01  -5.24621044e-01
   4.99500273e-01  -1.37775256e-01   1.45831658e-01   1.35785542e-01
   5.30012485e-02   4.58792781e-01  -3.65681617e-01  -2.13217102e-01
   4.32014874e-01  -1.61548625e-01   3.44794348e-01  -3.77503402e-02
  -7.75463214e-01  -5.95965636e-01  -3.08052154e-02  -1.78638052e-01
   3.29416029e-01   2.75569972e-01   5.38903799e-02  -2.06994272e-01
  -4.03068489e-01   1.96631352e-01   5.71284010e-01  -8.19141975e-01
   4.87334732e-01   3.26568989e-01  -1.86589816e-01   5.74410510e-01
  -8.65742691e-01   2.45227509e-01  -3.47736923e-01  -3.81285010e-01
  -4.28176823e-01  -4.65620303e-02  -8.35228944e-02  -2.63173282e-01
   7.70698794e-02   3.82203854e-01  -1.16713122e-01   3.20327813e-01
   1.70390759e-01   2.18415009e-01  -1.01948263e+00  -6.62644052e-01
   2.00597895e-01   1.55239658e-01  -2.17157198e-02  -2.45809996e-01
  -2.75015736e-01  -4.70479980e-02  -2.27912028e-02  -3.94918116e-01
   9.44659156e-02   2.29689278e-01   9.24637834e-02  -2.69701021e-01
   4.23932815e-01   1.54953277e-02  -6.35880292e-02  -2.59638434e-01
  -8.56235925e-04  -2.76353088e-01  -1.64925506e-01  -2.40244329e-01
  -4.25712883e-01  -2.54126288e-01   2.66295649e-01  -3.11255740e-01
   3.39134164e-01  -1.09447829e-01   3.82366941e-01   2.20036338e-01
   1.07592012e-01   4.00763552e-01  -1.40467599e-01  -1.37533981e-02
  -1.06988868e+00   2.95496141e-01  -8.11026317e-02   4.52159415e-02
  -2.78069223e-01  -5.95785389e-01  -3.41885734e-02   1.81172771e-01
  -5.42159933e-01   6.11280122e-01   2.18129214e-01  -9.52857354e-02
  -4.22137585e-01   3.81007473e-01  -4.31345216e-01   6.45393918e-02
   1.41692083e-01  -6.66907885e-02   4.18078846e-02  -1.52618191e-01
  -7.93934202e-02  -5.82499831e-01  -1.87694048e-02  -1.03751346e+00
  -2.64732991e-01  -3.61546252e-01   9.13774656e-02  -1.52564320e-01
   4.49515024e-01  -7.92294236e-01   3.40229857e-01   3.44081676e-01
   2.07166091e-01  -2.00005996e-01   1.84919478e-01   3.93009427e-01
   5.40721259e-01  -6.36739108e-02  -1.01074266e-01   2.45548983e-01
  -7.34547248e-01   3.85795459e-01   2.90510966e-01  -7.44243587e-01
  -3.65932722e-02  -1.28658694e-01  -1.41699676e-01  -6.56663288e-01
  -2.82833078e-01   3.65902768e-01   7.13942671e-01  -7.96642821e-02
  -1.37122338e-01   1.54711621e-01  -3.67214446e-01   4.60890510e-01
  -3.67951650e-01   3.20145721e-01  -4.37471451e-01  -1.52152040e-01
  -6.25869066e-02   3.02932433e-01  -5.14870905e-01  -6.26170972e-01
   4.13190509e-01  -1.02250334e-01  -3.66815397e-01  -3.94967073e-01
  -2.66406892e-01   2.15288265e-01   6.15851318e-01   5.63872372e-01
  -8.89418652e-01  -3.67878289e-01   2.02068658e-01   6.46667610e-02
   1.49284159e-02   6.37928577e-02  -2.05625440e-02   7.61227046e-01
  -3.29080069e-01  -1.72043720e-01   4.51831204e-01   2.90236159e-01
   2.50886386e-01   1.87000500e-01   2.02327878e-01  -2.33423422e-01
  -5.17710171e-01   7.46086691e-02   3.53238428e-01  -5.73924074e-02
  -4.36656565e-01  -3.30983224e-02   1.98880739e-01   1.62148838e-01
  -4.05027721e-01  -4.75746709e-02  -2.52676264e-01   5.08630183e-01
  -1.22951423e-01   4.04175114e-03  -3.72535548e-02  -2.35689717e-01
   5.66978404e-01  -1.00828052e-01   2.05728243e-01  -2.04538304e-01
   2.41157018e-01   2.42345041e-01   2.10546149e-01   1.34108038e-01
  -1.04948337e-01   2.53277137e-01  -1.45491852e-01  -4.56002065e-01
  -3.57476323e-01   2.43917653e-01   4.05429200e-01  -1.09762924e-02
  -2.95726785e-01  -1.59407473e-01  -5.43368366e-01   4.70453807e-01
  -1.28249243e-01  -4.59706970e-02  -2.30811423e-01  -1.65770809e-01
  -1.75468900e-01  -2.44430445e-01   2.20297922e-01   5.11963936e-02
  -2.35587561e-01   1.46070808e-02   4.95560573e-01   4.40133021e-03
   3.38128987e-01   3.90208204e-01  -3.66547689e-01  -3.55022292e-01
  -1.77572524e-01   2.72930498e-02  -1.90124341e-01   6.06278260e-01
  -7.12334717e-02   1.05381559e-01  -3.83705785e-01   2.54959502e-01
  -4.16026983e-01  -2.68862513e-02  -1.21480501e-01  -2.48124139e-01
  -2.68773981e-01   4.01582757e-01   2.09199347e-01   4.38944228e-01
  -9.50289727e-02  -6.30821081e-01  -1.94982541e-01   1.21620262e-01
  -2.05433602e-01  -3.04993912e-01   3.21822272e-02   4.53306996e-01
  -4.66215925e-01   2.69294303e-01  -1.63161381e-01   8.84142476e-01
  -3.00708722e-01   8.18861327e-01  -1.79523557e-01  -1.88369218e-01
  -2.21725620e-01   1.26363834e-01   2.27521928e-01  -6.77844505e-02
  -8.99829857e-02  -1.51942202e-02  -4.74782128e-01  -1.72990105e-01
   5.37971252e-02   7.06403216e-02   4.27140325e-01  -4.10864780e-01
  -4.67279658e-01  -9.92361147e-02   2.39058990e-01   4.35892134e-01
  -6.03435890e-02   1.11579639e+00   5.32438470e-01   1.12279378e-01
  -1.31022087e-01  -3.84304138e-01   5.22710932e-01  -5.39305212e-01
  -6.28436637e-02  -5.88000428e-02  -1.95143255e-01   5.38817048e-01
  -5.68579409e-01   5.25259845e-01   3.75690997e-01  -1.37164319e-01
  -3.87417261e-02   4.47448239e-01  -1.42277612e-01   2.48352953e-01
  -9.49813645e-03   2.04720447e-01  -1.30165676e-01   2.21459022e-01
  -5.83997089e-02  -3.34486868e-01  -4.85419696e-02  -2.38053301e-01
   1.17817007e-01   6.15446520e-02   3.58081064e-01   8.84299339e-02
  -1.54966627e-01  -6.10821886e-01  -1.86428396e-01   3.99867122e-01
  -1.55960093e-01   6.57549468e-01  -1.24627401e-01  -4.62506469e-01
  -4.24527337e-01  -2.18707728e-01  -1.80464836e-01  -1.97730864e-02]

2.1.3. Compute the mean-of-means, it's standard deviation, and 95% confidence region.

In [67]:
# Mean of means calculations
mu = np.mean(m)
print("Mean of means:",mu)

# Standard deviation calculation
sd = np.std(m)
print("\nStandard Deviation of means:",sd)

# Calculating the confidence interval
CI_Upper = mu + (1.96 * sd)
CI_Lower = mu - (1.96 * sd)
print("\nCalculating confidence interval using theoretical method")
print("95% CI Upper limit:",CI_Upper)
print("95% CI Lower limit:",CI_Lower)
# Calculating confidence interval using percentile method
print("\nCalculating confidence interval using percentile method")
print("95% CI Upper limit",np.percentile(m,97.5))
print("95% CI Lower limit",np.percentile(m,2.5))
Mean of means: -0.0083678422132

Standard Deviation of means: 0.325198259785

Calculating confidence interval using theoretical method
95% CI Upper limit: 0.629020746965
95% CI Lower limit: -0.645756431392

Calculating confidence interval using percentile method
95% CI Upper limit 0.574447592609
95% CI Lower limit -0.622311985883

2.1.4. Repeat the above with three different (and very different) N-s, such as 10, 1000, 100,000. Show how the mean-of-means, it's standard deviation, and the confidence region depends on N.

In [68]:
R = 1000
N = 100
In [69]:
m = np.random.normal(size=(R,N)).mean(axis=1)
print("Means:",m)
Means: [ -2.01859269e-02   9.80902819e-03   1.63381037e-01   1.18392676e-01
  -7.46031926e-02   2.50287219e-03   1.34452931e-01  -8.33828451e-02
  -9.21030321e-02   1.78868169e-01  -1.68029600e-01   1.29249179e-01
   1.68134657e-01  -1.15020927e-01  -5.52747925e-02   3.83709283e-02
  -1.33981163e-02  -2.49139263e-02   1.13274989e-01  -5.64437388e-02
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  -4.58726829e-02   1.97939897e-02  -1.27197246e-01   9.19535353e-02
   9.72891504e-02   1.59090672e-01   1.02938334e-01  -4.79869923e-02
  -5.67197833e-03   8.94032322e-02  -7.56270410e-02   1.57019666e-01
   3.71148777e-02  -5.39980643e-02   2.17090060e-01  -8.45984670e-02
   1.68305088e-01  -7.16339295e-02  -2.75647118e-02   2.17092088e-02
  -1.81050391e-01  -7.73215556e-02   2.57207187e-02   7.18883045e-02
  -7.89306693e-02   9.27253779e-02   1.51299592e-01   2.45335311e-02
  -6.20727415e-02   5.61404043e-02  -6.32370943e-02  -2.00777841e-02
  -3.08300809e-02   5.02857560e-02   5.50068017e-02   2.25254565e-01
  -1.57544325e-02   3.78092643e-02  -1.14148706e-01  -5.46956162e-02
  -1.22177429e-01   2.46636421e-03   2.78059034e-01   8.51170964e-02
  -1.31179606e-01  -1.58788543e-02  -8.70326553e-02  -8.06108882e-02
  -5.54454036e-02   3.49341265e-02   4.65723019e-02   2.83295144e-02
   6.54180031e-02  -1.64426791e-02  -1.96531573e-01  -1.36340702e-02
  -1.63095133e-02  -1.35148454e-01   2.37746096e-02   5.30349618e-02
   1.60933519e-01  -1.22951025e-01   1.54309688e-01   1.23125337e-02
   5.21120364e-03   8.12698331e-02   1.63562379e-01  -9.84251556e-02
  -7.61709838e-02  -1.14583797e-01  -3.47043630e-02   8.46165706e-03
   9.87119210e-03  -7.97033114e-02  -7.98858225e-02   1.03566952e-01
   1.25669712e-01  -3.65011255e-02  -9.27704084e-02  -2.84178386e-02
   7.26148400e-02   1.37154281e-01   4.68027495e-02  -2.71919761e-02
  -7.18141096e-02  -9.89479757e-02  -4.43985868e-02  -5.50979081e-02
  -1.34540527e-01   3.37906531e-03   4.75228933e-02  -4.62193141e-03
  -1.33289941e-02  -2.54900970e-01   1.58482495e-01  -6.14281464e-02
  -1.81176951e-01   1.08143075e-01   1.56309640e-01   5.38475544e-03
  -1.00144191e-01  -1.90805924e-03  -6.11514373e-02  -1.03193188e-01
  -1.38217087e-01   1.37128531e-01  -1.24435002e-01  -5.27826927e-02
   2.74178053e-04  -6.82070704e-02  -7.28490718e-02   1.29046546e-01
  -1.04472073e-01   3.67580679e-02   1.94981554e-01  -5.92520860e-02
  -1.79072311e-02  -4.07211166e-02  -1.24652494e-01  -1.75786255e-01
   3.55464740e-02   1.21034493e-01   1.45726039e-01   1.86843171e-01
   1.18146863e-01  -1.25733766e-01   8.58287139e-02   4.85048235e-02
   3.07240217e-02   1.71234577e-01   2.10373197e-01  -1.83081152e-02
  -9.48038023e-02   1.48057598e-03   1.42045629e-01   2.39712977e-02
   1.12218480e-02   3.75100128e-02   1.20629374e-01  -1.42932220e-01
  -7.49373665e-02   8.56502901e-02   9.11240472e-02  -1.31990639e-01
  -8.16207747e-03   3.17955400e-02  -4.86910202e-02  -2.58844129e-03
  -9.53514440e-02   1.20714821e-02   3.13605666e-02   1.20993634e-01
  -6.49034092e-03   1.01771012e-01   4.09784152e-02   2.78317728e-03
   1.15348698e-02  -1.04720237e-01   1.46704763e-01  -9.75362392e-02
  -1.43252564e-01  -1.57980834e-01   1.26402778e-01   9.16543628e-02
   3.39308646e-02   1.36213481e-02   3.47148082e-02   2.55396944e-01
   1.47014490e-01   5.71406744e-02  -4.94250275e-02   7.88755604e-02
  -2.89367229e-02  -2.58645919e-02  -4.63822729e-02   9.10184388e-02
   6.16509811e-03  -1.98942832e-02   5.62329071e-02  -8.91610983e-02
   5.08299825e-02  -3.04768930e-01   6.23382657e-02   3.43753064e-02
   1.65551906e-02   1.04826408e-02  -1.83061545e-01   4.07125153e-02
  -1.22741474e-01   1.49524333e-01   7.58940606e-02  -3.00726171e-02
   2.23613271e-02  -1.25702834e-01  -1.22805095e-01  -7.69113037e-02
   4.04043693e-02   7.99747407e-02   2.84313817e-02   4.35012321e-02
  -1.63057019e-02   1.35376865e-01   1.13743202e-02  -8.75651163e-02
  -1.60542446e-02  -1.23259612e-01  -7.01928209e-02   7.39986757e-02
   4.44409346e-02  -1.01757640e-02  -6.10462756e-02   1.97462469e-01]
In [70]:
# Mean of means calculations
mu = np.mean(m)
print("Mean of means:",mu)

# Standard deviation calculation
sd = np.std(m)
print("\nStandard Deviation of means:",sd)

# Calculating the confidence interval
CI_Upper = mu + (1.96 * sd)
CI_Lower = mu - (1.96 * sd)
print("\nCalculating confidence interval using theoretical method")
print("95% CI Upper limit:",CI_Upper)
print("95% CI Lower limit:",CI_Lower)
# Calculating confidence interval using percentile method
print("\nCalculating confidence interval using percentile method")
print("95% CI Upper limit",np.percentile(m,97.5))
print("95% CI Lower limit",np.percentile(m,2.5))
Mean of means: 0.00190155515147

Standard Deviation of means: 0.100268834263

Calculating confidence interval using theoretical method
95% CI Upper limit: 0.198428470308
95% CI Lower limit: -0.194625360005

Calculating confidence interval using percentile method
95% CI Upper limit 0.190934187787
95% CI Lower limit -0.181710312345
In [71]:
R = 1000
N = 10000
In [72]:
m = np.random.normal(size=(R,N)).mean(axis=1)
print("Means:",m)
Means: [  1.28832216e-02  -7.81893634e-03  -8.41854650e-04   1.16463261e-03
   5.08899796e-03  -8.89135186e-03  -8.45943670e-03  -1.40132610e-02
   3.40346067e-03   8.99347671e-04  -4.48341660e-03   2.89229772e-02
   1.34621598e-02   3.82281940e-03   1.04396819e-02   2.85303019e-02
   5.38421325e-04   6.12880045e-04  -5.64853948e-04  -1.93797957e-02
  -1.24502181e-02   1.47391154e-02  -1.12331610e-02   1.54432694e-02
  -2.55335159e-03   4.14043006e-04   1.03032483e-02  -1.41343946e-02
   6.95557842e-03  -5.69663041e-03   3.24376469e-03  -8.23652694e-04
   7.25249520e-05   6.79222989e-03  -3.49576388e-03  -3.00555055e-03
  -1.19795138e-02   1.53522963e-03  -2.85566767e-03  -8.11857333e-03
  -1.42984905e-02  -1.92907426e-03  -8.25188825e-03   1.01455629e-02
  -1.40552306e-02   6.04639465e-03  -9.12356204e-03   1.21161936e-03
   3.06844785e-03  -1.48296911e-03   3.49530391e-03   8.87425556e-03
   2.99337207e-03   1.00365386e-02  -1.88056574e-02  -1.98205847e-02
  -3.86267486e-03  -3.64504916e-03   6.12773851e-03  -9.33103863e-03
   1.13040010e-02  -8.14768571e-03  -1.85191125e-03  -6.91413386e-04
   7.18527578e-03   1.13047987e-02   4.41625715e-03   1.12804849e-03
  -5.45308798e-03   2.17773914e-02   8.55900011e-03   4.38205976e-03
   6.14025727e-03  -2.45810177e-02  -1.81272756e-02   4.37896663e-03
   1.43131285e-02   2.58007746e-03  -9.03856590e-03   9.09245620e-04
  -3.67051976e-03  -1.25437544e-02  -1.38098895e-02   1.34769887e-02
   1.58917807e-02  -6.40111042e-03  -2.61551058e-03  -1.78324292e-03
   5.92102735e-04  -5.99687356e-03   1.02648443e-02   9.06156866e-03
   2.67142063e-02  -1.10855568e-02   2.54851410e-03  -7.09949779e-03
  -6.36306601e-03  -1.78524151e-02  -4.39649246e-03  -1.45873602e-03
  -3.27491229e-03   3.97370374e-03  -1.12017884e-02  -7.10354884e-03
   7.44846305e-03   1.08540555e-02  -6.20851564e-03   6.16997491e-03
  -1.14995153e-02   1.50430609e-02  -9.47377701e-03   4.61863573e-03
   8.66631360e-03   1.22731092e-02  -9.81141534e-03  -9.21800048e-03
   6.65142984e-03  -2.90501891e-03  -1.03165354e-02   1.22305017e-03
   2.03950620e-02  -2.06400288e-03  -1.26811025e-02  -9.18705621e-04
  -4.34578081e-03   1.45181338e-03   7.25367203e-03  -3.94794608e-03
   1.15422345e-02   4.97034976e-03   9.61302725e-03  -6.69741273e-04
  -1.17592128e-02   9.53871255e-03  -2.05361211e-03  -2.26420477e-02
   1.06634865e-02  -5.50824824e-03   1.18940527e-03   9.46802030e-03
  -2.53027671e-03  -1.61636615e-02  -1.00566654e-02   6.45370112e-04
  -9.02610977e-03   1.47906867e-03   1.22874656e-02  -1.68533245e-02
   1.06309331e-02   1.38500075e-02   9.82010808e-03   1.29414339e-02
  -1.05000603e-02   1.16450201e-03  -4.17232804e-03  -7.97120545e-03
   8.19770562e-03   4.38018212e-03  -2.86844291e-02   1.70487485e-02
  -9.04669382e-03   5.72478072e-03  -1.26881496e-03  -4.90618811e-04
   1.49216776e-02  -1.10791503e-02  -2.21323075e-03  -5.39365145e-03
  -1.16725446e-03   1.91107388e-02   1.70962815e-02   1.10466890e-02
  -1.94547915e-02   3.73912917e-04  -5.30734153e-03   2.88773270e-02
   7.39395572e-03   6.64989171e-03   6.75758835e-03  -4.50078199e-03
   3.13404829e-03   1.27598956e-02   4.50801194e-03  -4.97432524e-03
  -1.12831867e-02  -6.66020791e-03  -1.37297055e-02  -6.34858765e-03
   1.56240615e-02   4.07721637e-03   1.12985829e-02   1.26587878e-02
  -1.91421355e-02  -1.54747519e-02  -8.68676305e-03   8.37943881e-03
   1.51081737e-02   2.18907845e-03  -4.01695676e-03   6.02856449e-03
  -3.17276108e-03   1.07317772e-02  -8.60213832e-03   1.89463870e-03
  -1.15710054e-02   1.01226793e-02   1.05226267e-02   2.39793636e-04
  -1.94547666e-03   7.55390856e-03   5.20620597e-03   3.87356352e-03
   5.44057839e-03  -1.83828961e-02  -4.25090775e-03  -3.19845792e-04
  -1.79020340e-02   5.78223949e-03  -3.27594616e-03  -3.00671190e-03
  -7.02353849e-03   1.45219309e-03   2.52192021e-04  -3.54427347e-04
  -4.25127063e-03   7.97341090e-04   8.20277776e-03  -2.76094272e-03
   4.33696538e-03  -6.22313396e-03   2.09586124e-02  -4.29804147e-03
  -8.59661992e-03  -5.09724547e-03  -2.24634569e-04   3.19666065e-03
  -3.15945736e-02  -1.43067661e-03  -9.14529353e-03   7.71643825e-03
   1.46208191e-02  -8.26449987e-03   1.04011514e-02  -5.22534404e-03
  -4.72660095e-03   1.15496381e-02  -5.53228548e-03  -1.79071529e-02
   3.19354205e-03  -4.35947588e-03   1.92619050e-02   4.19692832e-03
  -1.56000392e-02  -8.27125640e-03  -5.03277858e-04  -2.24212257e-02
   1.53552796e-02   7.61640987e-04  -2.74353876e-03  -1.65478276e-03
  -4.50872248e-03   2.50647827e-02   1.00566346e-02  -2.84393860e-03
  -2.49415893e-02   6.88745147e-03   3.18091586e-03   1.30113630e-02
  -1.59985109e-02   1.86173774e-02   9.23141968e-03   1.10694532e-02
   8.43863693e-03  -1.39304205e-02  -1.51678351e-04   3.05311316e-03
   7.08317118e-03   1.20403663e-03  -1.22721310e-02   2.39795820e-04
   9.48025662e-04   1.83883592e-03   4.59830029e-03  -6.81537506e-03
   5.03059031e-03   1.32312208e-02  -5.36736913e-03  -2.04914925e-03
   5.90483398e-03  -6.06681876e-04  -1.29974975e-02   6.88656198e-04
   7.99569199e-04  -5.38548234e-03  -1.15205738e-02  -8.69927499e-03
  -1.71684694e-02   1.17049208e-02   5.51279556e-03  -8.84131201e-03
  -1.11336326e-02   1.54186487e-02   1.73096369e-02  -3.54716735e-03
  -2.47904080e-03   7.18861395e-03   9.92489426e-03  -3.29141393e-03
   2.31923344e-03   2.88691511e-03   8.01459863e-03  -1.91101581e-02
  -9.51352905e-03   6.21358048e-03  -1.79684985e-02  -3.10771412e-03
   1.23562616e-03   1.50262613e-02   8.72756788e-04  -1.76473986e-02
  -1.69127467e-02  -5.73766604e-03  -1.23602268e-02  -9.86583329e-04
   1.83971437e-02   8.28000128e-03   1.33282232e-05  -1.86599615e-03
  -1.21893170e-02   1.35039036e-02   5.63905999e-03  -1.18710756e-02
   1.23951718e-02  -5.81380052e-03   6.77915700e-03  -1.62056622e-02
  -4.20331601e-03   1.73276697e-03   6.62882309e-03   4.21549042e-03
   6.39584765e-03  -1.33940849e-02   2.24806149e-02   4.06189287e-03
  -9.04249693e-03  -2.17442044e-02   5.73696322e-03  -1.59797341e-03
  -1.67947084e-02   3.23056896e-03   4.82070849e-03   6.82580548e-03
  -9.02380710e-04   1.08874945e-02  -1.05248970e-02   6.49773308e-03
   3.27517365e-04   7.79853927e-03   1.25317044e-02  -8.34576701e-03
  -1.66929284e-03  -5.20874949e-03  -6.51786410e-03  -1.04913844e-02
  -1.22663235e-02   6.51254137e-03  -4.08155984e-05  -1.08080973e-02
   1.99522522e-02  -2.94106884e-03  -1.25001993e-02   1.53758483e-02
   2.07363742e-03  -1.91614573e-02   7.79483482e-03   8.69158884e-03
  -1.00645890e-02   2.78657133e-03  -1.88612545e-02   5.22340514e-03
  -3.66973044e-03   1.66736864e-02  -1.59984984e-02   8.95714029e-03
  -1.71685321e-03  -7.01717337e-03  -8.27218959e-03  -1.41106726e-02
  -1.03484670e-03  -7.94835367e-03   1.08645845e-02  -6.90869341e-04
   8.78600172e-03   1.39627634e-02   9.54770519e-03   2.27324218e-03
  -5.44337258e-03  -1.01004450e-02  -8.60249868e-03   1.21076398e-02
   1.50820912e-02   1.08007162e-02   1.50886837e-02  -1.64571953e-02
  -2.28245506e-02   2.31743132e-02  -9.13344166e-03  -8.93872916e-03
   3.37982398e-03  -1.38623006e-02  -6.56834201e-03  -2.20380692e-03
   1.40785047e-02  -5.72693620e-03   9.80234346e-03  -4.67620975e-03
   8.07366560e-03   2.84273487e-03   7.06944565e-04  -2.26472080e-04
   2.95924595e-02   4.43366483e-03   3.38918143e-03  -1.70519385e-02
  -2.48307031e-03  -1.21939875e-02  -6.97012946e-03  -2.99379587e-03
   3.21977664e-03  -1.58234501e-04   5.51532654e-04   6.31092608e-03
  -6.87491555e-03   9.68363087e-03  -1.14656900e-02  -1.47788305e-03
  -7.39103912e-04  -5.64030834e-03   6.49948198e-03   1.88023377e-02
  -6.62973535e-03  -2.84648126e-03  -8.57143621e-03   1.43057988e-02
   4.85462662e-03   1.06050626e-03  -1.28476864e-02  -1.78804056e-03
   2.52122917e-03  -3.36253286e-03  -1.41287862e-02  -1.49223915e-03
  -1.70405316e-02   1.50523757e-02  -1.27592352e-02   7.70120914e-03
  -1.98994624e-04   4.46157188e-04   3.57354778e-03   3.47102096e-03
   2.32261357e-03  -5.84413807e-03   1.47108542e-02  -3.89979258e-03
   9.36338061e-03   9.60571991e-03   1.03862079e-02   9.05062882e-03
   9.45544713e-03   1.56092640e-03   1.21111871e-03  -5.57342601e-03
   1.39983452e-02  -6.67494386e-03  -7.65151376e-03   4.61490699e-03
  -9.92644678e-03   1.34584955e-02  -5.81781723e-03   5.58581944e-03
  -7.14248759e-03  -3.04705809e-03  -1.17112938e-02  -7.99907322e-03
   5.28176723e-03   2.76203882e-02  -8.82858678e-03  -7.61102835e-03
  -3.15835900e-03   4.39077348e-03   2.34509390e-02   1.28013301e-02
  -1.38031849e-03   4.21804822e-03   9.39984734e-03   2.69404066e-03
   3.18230580e-02   1.90231851e-03   7.29870653e-03   7.22946809e-03
   1.33341802e-02  -9.88448200e-04   4.45213452e-03   5.05949130e-03
   1.81599369e-02  -1.20421422e-02   5.12932383e-03   1.80350857e-02
  -5.86468659e-04   8.24037783e-03   1.15315019e-02   1.14977847e-03
  -5.59823509e-03   5.16874516e-04   4.26996719e-03  -1.08689338e-02
  -1.98445709e-02   8.03088417e-03  -3.05081631e-03  -6.17436185e-03
   1.49695994e-02   3.00937420e-03   3.70041929e-03  -5.62264516e-03
  -4.75023412e-03  -8.02419836e-03  -5.17661468e-03   6.66001292e-03
   6.40965300e-03  -1.57126673e-02  -7.98729110e-03   9.00921531e-03
  -1.53186480e-03  -2.24488304e-03  -7.95377446e-04   9.59972225e-03
  -4.65893178e-03  -4.02966416e-03  -8.37279712e-03   3.63168499e-04
   2.02821961e-02  -3.74324558e-03  -1.23364252e-02   8.30466032e-03
  -1.12060700e-03  -3.89970404e-03   5.65275539e-04   4.94683115e-03
  -8.19288502e-03  -1.99034646e-02  -1.13567653e-02   4.25899304e-03
  -2.74083557e-03   2.89364110e-03  -1.20249870e-02   6.33927806e-03
   1.74042699e-02  -6.35849837e-03   1.03357465e-02   5.57332333e-03
  -2.89165158e-03  -1.26845452e-02   8.60780400e-03   4.02521995e-03
  -1.60764943e-02   3.85252303e-03  -4.03398253e-03  -2.98315652e-03
  -3.10810476e-03   1.86923326e-02  -9.92186775e-03   6.81014944e-04
   6.56676920e-03  -2.43522708e-02  -4.26854174e-03  -1.05594261e-02
  -5.54019142e-03   1.17688521e-03   2.60246173e-03   2.93993156e-03
   1.71731056e-02   2.69522138e-03  -5.80082178e-03   5.41501829e-04
  -3.81844995e-03   3.44057126e-03   1.45315106e-02   1.45234712e-03
   8.91481371e-03   3.31404988e-03  -2.35541973e-02  -1.16844606e-02
   1.16277614e-02  -2.68992895e-03  -2.17785077e-04  -1.03333923e-02
   1.56664016e-03  -1.95435246e-03  -7.70418152e-03  -6.31205185e-04
  -1.86014561e-02  -5.57101633e-03   1.25801242e-02  -6.86732756e-03
  -1.82554730e-02  -7.72073688e-03   1.52094650e-03  -8.47386536e-03
   5.24194065e-03  -9.54539647e-03   2.43876034e-02   3.22971563e-03
  -7.64530109e-03  -1.80674678e-02   1.27991491e-02  -9.22183756e-03
  -8.36683094e-03   3.03161244e-03  -3.15959864e-03   1.17405889e-02
   4.70619766e-03   9.19879891e-03  -4.98263987e-03  -1.53019943e-02
  -4.17374460e-03   5.46015120e-03  -1.12388157e-02  -9.59685090e-03
   3.93306848e-03  -2.17842236e-03   2.83345182e-03   3.09496218e-03
  -4.09503233e-03  -3.47698131e-03  -1.35073486e-02  -1.34155375e-02
   5.13923996e-03  -2.41987925e-03  -1.98482894e-02  -8.91600279e-04
   1.57236646e-03  -5.67329933e-05   2.94178391e-03  -2.19069979e-03
   4.15614685e-03   2.09064389e-02   1.11046905e-02  -3.35786016e-03
  -1.57543838e-02   1.54310926e-03   8.96274300e-03  -3.11335690e-03
   5.55967574e-03   7.46384434e-03   3.03087876e-03  -1.03067565e-02
   2.05217679e-02   7.05589302e-03   2.05037124e-03  -8.47937303e-03
   3.46732321e-03  -5.33566555e-03  -1.81783738e-02   2.02824779e-02
   1.48433059e-03   1.44319375e-03  -1.94934182e-02   5.28762868e-03
   5.00046709e-03  -1.18588670e-03   2.37246445e-03   3.17303391e-03
  -1.99672990e-02  -8.04093205e-03   9.90867775e-03   7.61336724e-03
   9.93534993e-03   7.67711464e-03   6.60087649e-03  -9.81995871e-03
  -1.96734335e-02  -2.67645705e-03   7.08695941e-03  -1.48269843e-02
   1.39083316e-02  -1.01755841e-02   1.45499973e-02  -1.87611900e-03
   6.72708598e-03  -9.92515174e-03  -1.25389120e-02   1.43917642e-02
   8.75190021e-03   4.17303735e-03   2.02068476e-02   6.26881334e-03
   1.43943346e-03  -2.17526430e-03   4.25487563e-03   4.59673554e-03
   2.10197418e-03   4.67421712e-03   6.78408994e-04  -3.52337949e-03
  -3.20844286e-04  -2.32983012e-02  -6.54761975e-03   9.16104820e-03
  -4.44676234e-04   9.00651765e-04  -6.04282485e-03   2.48593720e-03
   1.18930132e-02  -2.43795849e-03   2.08465401e-03  -3.03597559e-03
   4.56929039e-03  -1.61795099e-02  -1.19075244e-02   7.88782566e-03
  -1.19772105e-02  -1.12855608e-02  -5.55201606e-03  -3.16955994e-03
  -8.79795475e-03  -2.88766237e-03   4.05071964e-03   4.66571409e-03
   1.82172282e-04  -6.82341918e-03  -4.19319403e-03   4.66815822e-03
   1.20140504e-03   7.08364224e-03  -1.01158351e-02  -1.63410845e-02
   3.65936370e-03  -7.30693462e-03  -1.02890665e-02  -6.39221143e-03
  -2.09178820e-05  -4.71358211e-04   1.52882929e-02   4.64489627e-03
  -1.94394438e-03  -1.17563484e-04   2.93148896e-03  -4.24240383e-03
   2.08758016e-04  -1.34028057e-02   1.19824884e-04   4.98941573e-03
   1.33529530e-02   8.10582391e-03  -1.65264439e-02   7.86171264e-03
  -9.42981760e-03   5.39785349e-03  -1.07867684e-02   3.06600970e-03
  -2.58346336e-03  -6.39617671e-03  -1.18115081e-03  -2.10955367e-03
   3.43577341e-03   1.35381634e-02  -1.92335650e-02  -1.94849008e-03
  -7.53046897e-03   1.28896260e-02  -6.88373429e-03   7.07436028e-03
   1.30333527e-02  -6.67052879e-03   1.20103975e-03  -7.67601660e-03
   6.41608775e-03  -3.08620238e-03  -6.04070084e-03   1.40240729e-02
  -7.69534081e-03   7.75895447e-03   2.71818222e-05  -2.61231302e-03
   5.79781558e-03  -6.22816774e-03  -4.58120988e-03   8.29973833e-03
   2.01002615e-03  -2.68830599e-04  -1.06765774e-02  -3.97781993e-03
   9.68903388e-03  -1.99933199e-03  -9.34145426e-05   5.31300860e-04
  -4.60513983e-03   6.98991284e-03   2.45801578e-02   1.97083363e-02
   1.21098412e-02   8.59098982e-03  -8.19927155e-03  -4.45895007e-03
   1.45914111e-02   1.34271779e-03  -1.41569146e-02  -3.36582681e-03
  -1.81609685e-03  -1.05750017e-02   1.74026362e-03  -1.22045494e-04
   6.47498863e-03  -1.43236603e-02  -5.55992737e-03   1.52025349e-02
   4.17521784e-03  -4.94553125e-03  -3.70815071e-04  -9.27828255e-03
   3.29389942e-03   4.43272103e-03  -1.50060668e-02  -1.15423804e-03
   6.86467272e-03   9.55897861e-03  -1.32990257e-03   1.57263689e-02
  -1.41484148e-03   1.35613004e-02   1.56203555e-02   8.48212561e-03
  -5.72619330e-03   1.07447665e-03  -7.09453941e-03  -1.91014892e-02
  -1.78183296e-02   8.61190968e-04   2.64850413e-03  -4.56543162e-03
  -8.98781270e-03  -1.58113380e-02   1.17384349e-02   2.56150200e-03
  -1.33721644e-02   1.53823980e-02   2.43041207e-02   8.46033936e-03
   1.28303363e-02  -9.56491089e-03  -9.54723031e-03   6.12322995e-03
  -6.53565902e-03   4.16441890e-03  -2.24119445e-03   2.35516879e-03
   1.05377865e-02  -9.95265473e-03  -8.40476137e-03  -1.72830109e-02
   1.01031535e-02   5.47570816e-03  -4.65619414e-03   8.21057244e-03
  -1.11655907e-03   1.64531757e-04  -2.30161739e-03   4.33440658e-03
  -8.14442072e-03  -1.55349393e-03   1.26636307e-02   1.83519217e-02
  -2.14975314e-03   3.45586153e-03  -1.48362620e-03  -1.09085699e-02
  -1.62868551e-03   2.61532605e-02   1.68440028e-03   6.66765988e-03
   1.39284036e-02  -1.15202813e-02   5.64539936e-03   7.90644251e-03
   5.51928889e-03  -3.46051988e-03  -2.16533634e-02  -4.03897086e-03
  -1.10124837e-02  -1.49216536e-02  -6.75652388e-03   4.49165928e-04
   2.37145229e-03   9.09124491e-03  -9.84312582e-03   5.23453162e-03
  -1.31055033e-02   1.20802796e-02   9.11879503e-06  -7.19792566e-03
   4.55030648e-04   2.56774101e-03  -3.19100742e-03   1.04924399e-02
  -7.48786390e-03   4.95376789e-03  -6.98829331e-03  -4.80655612e-03
   1.30048635e-02   1.35218517e-02   6.81046514e-03  -4.45099042e-03
   1.11962644e-02  -3.65131382e-02   3.25530094e-03   8.48763973e-03
  -2.61079400e-03  -1.70521750e-03   1.25444263e-02  -1.66861386e-02
  -2.12596741e-02  -4.14088848e-03  -6.19418557e-03   1.06250664e-02
   4.74412068e-03  -1.72996894e-02  -7.10568610e-03  -2.26768069e-03
  -4.95450566e-03   3.00056741e-03  -2.07620368e-04   6.98172296e-03
  -7.07121284e-03  -5.61266153e-03   1.56653283e-02   9.18902548e-03
   1.36989922e-02  -2.16985201e-02   1.43629148e-02   3.70657436e-03
   4.43972240e-03  -7.84232021e-03  -2.18863773e-02  -7.42064801e-03
   4.29048829e-03   2.17658551e-03   6.11793778e-03  -1.10698841e-02
   6.92529631e-03  -5.92205813e-03   4.73690544e-03   6.74493214e-03
   1.12805726e-02  -2.32208052e-03  -1.68155988e-02   2.62330695e-03
   8.29046214e-03   8.23531258e-03  -2.84916612e-03  -5.91424087e-04
  -3.98549413e-03   1.00622679e-02  -5.40378012e-06   7.61537361e-03
  -7.61546711e-03   1.36218719e-02  -5.86547676e-03  -1.77696214e-03
  -2.11550477e-03   1.43598409e-02   1.27034984e-02  -5.51054849e-03
   6.78336900e-03  -2.12501304e-02  -6.48341453e-03   4.11746201e-03
  -9.00113221e-05  -2.16694058e-02   9.04887682e-04  -5.34587962e-03
   6.93489799e-03   2.19281034e-02  -5.25014489e-03  -1.16756626e-02
   1.09146252e-02  -1.50699220e-02   2.94815072e-04  -1.14813678e-02
  -1.82491869e-02  -9.40301063e-03   5.78058581e-03   7.04979010e-03
  -4.00292575e-03  -3.27360118e-03  -4.02892077e-03  -5.27658256e-03
   1.15451169e-02  -6.50208800e-04   1.32454984e-03   1.14400040e-02]
In [73]:
# Mean of means calculations
mu = np.mean(m)
print("Mean of means:",mu)

# Standard deviation calculation
sd = np.std(m)
print("\nStandard Deviation of means:",sd)

# Calculating the confidence interval
CI_Upper = mu + (1.96 * sd)
CI_Lower = mu - (1.96 * sd)
print("\nCalculating confidence interval using theoretical method")
print("95% CI Upper limit:",CI_Upper)
print("95% CI Lower limit:",CI_Lower)
# Calculating confidence interval using percentile method
print("\nCalculating confidence interval using percentile method")
print("95% CI Upper limit",np.percentile(m,97.5))
print("95% CI Lower limit",np.percentile(m,2.5))
Mean of means: -3.91945233643e-05

Standard Deviation of means: 0.00996623840045

Calculating confidence interval using theoretical method
95% CI Upper limit: 0.0194946327415
95% CI Lower limit: -0.0195730217883

Calculating confidence interval using percentile method
95% CI Upper limit 0.0197144341544
95% CI Lower limit -0.0194557571988

Based on the above results, we have following observations: a) N = 10, the Mean of means is 0.0055, Standard Deviation of means is 0.3158 approx. and the 95% confidence interval is 95% {0.6364, -0.6253} b) N = 100, the Mean of means is -0.00047 and Standard Deviation of means is approx. 0.1005 and the 95% confidence interval is {0.2009, 0.1892} c) N = 10000, the Mean of means is -0.0001, Standard Deviation of means: 0.010 approx. and the 95% confidence interval is {0.0192, -0.0188}

Hence, we can conclude that with increase in total number of observations, the mean reduces, standard deviation also decreases and the confidence interval becomes narrower.

2.2.1. Create a large number of pareto random variables. Show their distribution on a histogram. Attempt to make the histogram look good.

In [74]:
b = np.random.pareto(a=1,size=(10000)) + 1
plt.figure(figsize=(10,7))
plt.hist(b,log=True)
plt.show()

2.2.2. Pick R and three N values

In [75]:
R = 1000
N = 10

2.2.3. Create R times a sample of N Pareto values, and take a mean of those.

In [76]:
b = np.random.pareto(a=1,size=(R,N)) + 1
plt.figure(figsize=(10,7))
plt.hist(b,log=True)
plt.show()

2.2.4. Compute the mean-of-the-means, it's standard deviation, and 95% confidence region.

In [77]:
# Calculating means
mu = np.random.pareto(a=1,size=(R,N)).mean(axis=1)
print("Means:",mu)
Means: [  7.22963609e+00   2.31995914e+01   4.45448338e+00   3.35646710e+00
   3.70091196e+00   2.76940319e+00   1.82164212e+00   3.30799476e-01
   3.77709736e+00   1.81850342e+00   2.10615510e+00   2.10784854e+00
   1.21181012e+00   5.93580149e+00   4.86544330e+00   1.47668176e+01
   1.54847860e+00   1.91731790e+01   1.85445723e+00   2.08967063e+00
   2.12952518e+00   3.17867930e+00   1.43047987e+00   2.92002449e+00
   2.53538924e+00   4.58507231e+01   5.54783268e+00   2.41148021e+00
   6.85412200e-01   6.19850657e+00   2.96590115e+00   1.20169056e+00
   3.88418371e+00   1.53760494e+00   3.95578137e+00   1.05511414e+00
   2.20625142e+00   2.21278210e+00   1.65011860e+00   3.54616253e+00
   2.80526033e+00   1.66817388e+00   2.56894811e+00   1.01306669e+00
   1.18487762e+00   3.72338945e+00   2.24255975e+00   3.71409254e+00
   1.07563568e+00   7.40501923e-01   1.31335771e+00   2.17525218e+00
   1.99332352e+01   1.60158268e+00   1.79317702e+02   5.72597056e+01
   2.79630643e+00   7.19077800e+00   2.10303750e+01   2.43049148e+00
   2.96787956e+00   6.30585081e+00   8.83946916e-01   1.24491045e+01
   4.67924528e+00   1.06553423e+01   2.07645449e+00   1.23828619e+00
   4.69452902e+00   7.61247219e+00   9.93920893e+00   6.96021828e+00
   4.52008056e+00   5.93727351e+00   6.58391900e+00   1.64244871e+00
   9.12118857e+00   4.01984864e+00   1.70756540e+00   3.17380807e+00
   5.10047215e+00   1.94315656e+00   6.51014720e+01   1.96166447e+00
   4.02997921e+00   8.56695574e+00   2.69550759e+00   1.72579745e+00
   3.79388629e+00   6.65081589e+00   1.30676494e+01   4.31716013e+00
   2.48790003e+00   1.73274357e+00   2.50145060e+02   5.61416929e+00
   2.44290084e+00   1.03340937e+00   8.17539023e-01   4.88914760e+02
   8.67022150e+00   6.23728297e+00   1.69649351e+01   9.51171192e-01
   3.36423750e+00   3.67546142e+00   3.31389871e+00   1.33753689e+00
   5.20470346e+00   7.38057523e+00   1.85482442e+00   2.10534044e+02
   1.36627724e+00   3.08431036e+00   4.72841590e+00   1.42024736e+00
   8.56748829e+00   1.99064586e+00   1.47295044e+00   8.41226008e+00
   1.07175770e+00   2.16044402e+00   4.98355915e+01   3.57480928e+00
   4.62032878e+00   2.68010204e+00   1.23431654e+01   1.07262865e+00
   1.21110008e+01   8.57549859e-01   5.59295148e+00   4.18310988e+00
   6.15728258e-01   1.48891139e+00   1.30352839e+01   7.78165668e-01
   2.64800247e+00   7.27625982e+00   1.38434931e+00   1.38928623e+01
   5.46149059e-01   3.07182669e+00   1.67293035e+00   3.25939879e+00
   2.23439232e+00   1.32052889e+00   1.82624292e+00   4.10257093e+00
   1.74563229e+00   9.28644356e-01   6.20008263e+00   2.03989915e+00
   1.84166031e+00   6.84109798e+00   1.56151665e+00   9.37012086e-01
   8.16087691e+00   2.88705899e+00   7.87928540e+01   6.78509677e-01
   4.01327319e+00   4.84985976e+00   3.86473090e+00   8.11339212e+00
   4.96459442e+00   5.47001233e+00   3.27978978e+00   2.04091026e+01
   1.92782841e+00   3.87329100e+00   3.89859312e+00   1.92006208e+01
   3.37947209e+00   3.56782460e+00   1.95224477e+00   2.42548445e+00
   1.79024145e+00   1.00885611e+00   3.63508954e+00   1.67591874e+00
   3.61927405e+00   2.22909283e+00   4.59578622e+00   8.76009487e-01
   2.32697506e+00   2.69006182e+00   3.08558052e+00   1.93523167e+00
   2.03131972e+00   1.69384400e+00   8.93358119e+00   1.58829901e+00
   1.61219501e+00   1.32485977e+01   2.92200535e+00   8.23932498e+00
   6.31045559e-01   2.54286143e+00   3.23121672e+00   3.25480919e+01
   3.99468935e+00   1.46135912e+00   2.80735286e+00   2.91488928e+00
   2.63507234e+00   2.56966634e+00   1.06578643e+00   1.46147240e+00
   2.14335291e+00   8.66880835e+00   4.02800871e+00   4.15652650e+00
   3.78307112e+00   1.11129411e+00   1.58859998e+01   2.14057463e+00
   1.39339842e+01   1.32646864e+00   2.52770796e+00   1.34818967e+01
   2.61416293e+00   2.51693291e+00   2.45548079e+00   1.28974776e+00
   4.18467833e+00   2.53345790e+00   3.15381445e+00   4.24582892e+00
   7.33614999e-01   5.42389859e+00   1.05648941e+01   3.81393950e+00
   3.62111808e+00   1.98563157e+01   1.08419485e+01   3.97471366e+00
   1.24937858e+01   3.25517729e+00   2.69216826e+00   7.68414023e+00
   1.32335663e+00   2.50119290e+00   7.86997000e-01   6.23879110e+00
   4.29284290e+00   6.70514288e+00   8.54202908e+00   1.46219812e+00
   3.58748893e+00   7.08445470e+00   2.52707911e+00   5.90924518e+00
   5.31287520e+00   2.80547264e+01   3.51832725e+01   1.13895222e+00
   9.84225947e-01   1.15106759e+00   1.41942705e+00   9.45659411e+00
   5.12282922e+01   9.56351668e+00   1.26007998e+01   2.23529724e+00
   3.34174714e+00   6.09250309e+00   1.20911582e+00   3.94577564e+02
   2.44169716e+00   2.23872247e+00   5.21967312e+00   7.82711612e-01
   4.89182868e+00   1.88254938e+00   1.31172705e+00   2.60338654e+00
   2.06035369e+00   1.19703336e+00   2.94104137e+00   4.14124003e+00
   1.38044132e+00   2.72215623e+00   7.35826184e+00   2.19523830e+00
   1.98485803e+00   3.95734302e+00   1.65427215e+00   2.44037554e+00
   1.75039379e+00   1.94193565e+00   2.37039545e+00   6.28657175e+00
   3.70864676e+00   6.23936817e-01   1.26713425e+00   1.87602058e+00
   3.17577620e+00   7.08746129e+00   2.06011002e+00   3.69734775e+00
   1.72168924e+01   4.08950022e+00   5.58136343e+00   6.09511513e-01
   2.36991247e+00   1.85088634e+01   3.67520210e+00   1.81147584e+00
   5.75989750e+00   1.40598056e+03   4.66098152e+00   1.71825530e+00
   4.46358792e+01   7.21106947e-01   2.29907458e+00   4.44132753e+00
   8.77233766e+00   3.09118775e+00   2.22815214e+00   2.28382918e+00
   3.12015968e+00   3.19207507e+00   9.18528657e+00   7.15184711e+00
   2.56813366e+00   5.83706251e+00   2.01153043e+00   3.19382786e+00
   4.25835686e+00   1.28862450e+00   2.84359474e+00   3.23096682e+00
   9.20322544e-01   2.69651806e+00   4.17069557e+00   2.65340934e+00
   3.63904863e+00   1.81132673e+01   2.16320304e+00   1.92525136e+00
   1.50440451e+00   1.98600530e+00   2.18056243e+00   7.93455463e+00
   1.79712717e+00   1.52153179e+00   8.34744957e-01   1.33149645e+00
   8.63112365e-01   2.01596840e+02   4.85355293e+00   2.30614534e+00
   3.58844012e+00   1.96648068e+00   5.38252701e-01   2.22914887e+00
   3.56850773e+00   2.40473595e+01   1.06100236e+01   4.02053546e+01
   2.65771821e+00   2.49522611e+00   8.86905896e+00   4.94635702e+01
   1.70054962e+00   1.10383694e+00   2.89358966e+00   4.71106630e+00
   2.24747129e+00   2.97144462e+00   2.01446125e+00   7.24388318e+00
   4.91885201e+01   2.88468524e+00   5.01701871e+00   3.30311936e+00
   7.45315468e-01   4.66897120e-01   5.87972590e+00   1.30075418e+00
   1.68835176e+00   1.34681216e+01   3.00714814e+00   1.25729235e+01
   1.99864636e+00   5.81679405e+01   2.17740238e+00   2.32463070e+00
   2.82513149e+00   4.50146061e+00   2.29899937e+00   6.59300068e+00
   3.87715926e+00   1.00523171e+02   2.57930751e+00   5.92143193e+00
   8.96383914e+00   2.54108860e+00   4.05786775e+00   1.61297106e+02
   1.88168715e+00   4.55475907e-01   1.25270795e+00   1.18249066e+00
   1.08395403e+01   1.02430978e+00   3.12448916e+00   5.86675048e-01
   1.31206173e+00   3.51951637e+00   3.29580358e+00   5.98186240e+00
   2.42129416e+00   2.20807384e+00   1.95141652e+01   8.01394323e+00
   1.88582257e+00   1.95777732e+00   1.54738440e+00   2.43224883e+01
   1.37916634e+00   1.68327978e+00   1.16021913e+00   1.27054924e+01
   1.45001536e+00   2.46685453e+00   3.80028358e+00   1.31553603e+00
   1.52799489e+01   2.66486433e+00   1.79962913e+00   3.70112309e+00
   1.93840851e+00   1.31902945e+00   4.30796642e+00   1.39606188e+00
   4.76076230e+01   1.98982546e+00   2.15012708e+00   1.86361653e+00
   4.84063178e+00   1.81996702e+00   3.91702067e-01   1.46483703e+00
   9.01788791e-01   5.68022929e+01   2.39488174e+00   3.81205712e+00
   2.59197714e+00   8.14758525e+00   2.00113557e+00   1.09945959e+01
   5.33757562e+00   2.22168259e+00   3.53581818e+00   3.54201951e+00
   3.45733882e+00   3.88843483e+00   2.72676203e+00   3.89326772e+00
   1.03576903e+00   1.50091745e+00   1.43582802e+01   2.75599839e+01
   2.25392871e+00   4.43928325e+00   3.20863385e+00   2.48867816e+00
   1.67456327e+01   2.54729218e+01   1.67005677e+00   3.91038816e+00
   3.05837344e+00   4.40157381e+00   1.07232174e+00   4.09325401e+00
   2.12564146e+00   5.80153127e-01   1.65221499e+01   3.24642052e+00
   1.05532249e+00   1.80513886e+00   2.03037012e+01   1.10888615e+00
   7.13923809e+00   4.11541782e+00   5.07171486e+00   7.28258183e+00
   5.25680109e+00   1.53704348e+00   1.02579539e+00   2.63148645e+00
   2.55407393e+00   2.98288562e+00   2.00158690e+00   5.00467143e+00
   1.74903653e+00   5.86194163e+00   9.74016887e-01   2.03688241e+00
   9.56191690e-01   1.58939218e+00   1.12857199e+00   4.41021974e-01
   1.05158960e+00   8.28509844e+00   3.05899231e+00   4.95172502e+00
   2.10871219e+00   2.92930455e+00   1.48515051e+01   5.00373616e+00
   1.59476734e+00   4.38374091e+00   7.34948815e+00   9.11915576e-01
   1.85321870e+00   1.87273297e+00   1.78935057e+00   2.19156098e+00
   2.73752672e+00   2.14465196e+00   1.21175740e+00   4.27716396e+00
   1.20133179e+00   2.45437603e+00   1.33346157e+00   2.16192743e+00
   4.35429161e+00   1.14580335e+00   7.37822777e+00   3.99153412e+00
   1.77194131e+01   2.23993460e+00   2.56525222e+00   1.33410729e+00
   2.87679994e+00   3.26040566e+00   3.70809526e+00   7.07435035e-01
   1.59020047e+00   5.20212492e+00   2.54451807e+00   4.84434811e+00
   2.71449515e+00   2.25796169e+00   3.46305266e+00   1.89605118e+01
   1.52209598e+00   3.00205086e+00   2.20846732e+00   7.12899063e+00
   5.34208361e+00   1.86076542e+00   1.40368564e+00   3.67715142e+00
   6.16725054e+00   2.25440765e+00   4.11711074e+00   5.33714047e+00
   1.36591933e+00   4.65248076e+00   3.79305253e+00   2.87331832e+00
   1.12660858e+01   1.82760902e+00   7.19222100e+00   1.56057194e+00
   9.23061642e-01   2.67268869e+00   9.05744439e+00   5.01367575e+00
   1.48345173e+01   1.66956394e+00   4.93402955e+00   4.73229627e+01
   1.45505110e+00   1.02624851e+00   4.36975891e+01   7.53305380e-01
   4.57488895e+00   3.85050105e+00   1.64860952e+00   6.47710968e+00
   5.56656240e+00   1.35316607e+00   2.92038261e+00   1.06961315e+00
   3.01219891e+00   1.86399177e+00   2.94440200e+01   4.99457082e+00
   1.01865971e+00   1.72370509e+00   1.25002143e+00   9.15667823e+00
   8.83569201e+00   8.56402533e+00   2.15702924e+00   3.71717685e+00
   2.94719638e+00   2.59908533e+00   2.63325743e+01   9.53110489e-01
   7.16517801e-01   1.02072972e+00   1.78753013e+00   1.35907561e+00
   1.44767081e+00   3.94962620e+00   1.14220238e+00   3.19638473e+00
   4.32659302e+00   5.45767670e+00   3.03844529e+00   5.53926158e+00
   2.56698114e+00   3.00875919e+00   1.15980374e+01   6.57070592e+00
   7.13493248e+00   1.01315296e+00   1.70337898e+00   3.92902077e+00
   1.80265094e+00   6.23877953e+00   3.33427151e+00   3.86575048e+00
   9.96912573e-01   2.19092745e+00   1.74635954e+00   2.52509409e+01
   7.58322812e+00   1.28522108e+00   3.45606102e+00   4.52694604e+00
   5.14554474e+00   2.65452126e+00   1.74472943e+00   8.76914892e+00
   1.76066230e+00   1.99664074e+00   5.92459703e+00   1.17069504e+01
   2.36957207e+00   2.18321711e+00   7.67519055e+00   2.45564334e+00
   7.22231838e+00   5.76728803e+00   2.75761894e+00   2.57593174e+00
   1.71854588e+00   3.43474239e+00   3.67464635e+00   2.34805595e+00
   1.58227585e+00   7.23426449e+00   1.97468611e+00   4.68024545e+00
   2.08605559e+00   1.77606374e+00   7.92662014e+00   2.50198009e+00
   3.72728203e+00   3.20346275e+00   8.75877046e-01   3.37837880e+00
   9.30504376e+00   2.44038700e+00   6.01036732e+00   2.08113814e+00
   2.40741655e+00   2.26958234e+00   2.54162403e+00   2.89630624e+00
   1.74914705e+00   4.09331766e+00   2.19392451e+00   2.22279662e+00
   3.12804792e+00   1.74267509e+01   3.12924774e+00   2.53590507e+00
   1.88912325e+00   5.39834486e+00   5.07451237e+00   2.66853385e+00
   1.14878708e+00   2.78532562e+00   5.44714044e+00   1.56536585e+00
   1.12836050e+02   5.37626136e+00   2.98324908e+00   2.06005998e+00
   9.19247681e+00   1.55840108e+01   8.04187329e+00   3.31256971e+00
   2.51472673e+00   3.28186069e+00   1.90013203e+00   5.51783234e+00
   1.03780066e+01   2.65799194e+00   8.56903225e+00   2.07973461e+00
   4.86347637e+00   4.60425899e+00   3.96049062e+00   4.81080301e+00
   2.33960223e+00   1.83715429e+00   3.04238147e+01   1.93211672e+00
   3.08273482e+00   5.21241051e+00   3.99205956e+00   4.32826959e+00
   6.08167490e+00   2.30872337e+00   2.59020845e+00   3.18972941e+00
   3.33326784e+00   3.43002724e+00   5.31487447e+00   5.20506161e+00
   3.62729742e+01   3.11255025e+00   3.29337484e+00   6.27134359e+01
   6.52646787e+00   1.06878084e+00   3.94429562e+00   2.87400688e+00
   1.62526658e+00   2.33735516e+00   1.67365490e+00   2.30447849e+00
   1.39365043e+00   9.48119745e-01   1.07339914e+01   9.76575522e-01
   3.92565992e+00   8.84305808e-01   3.81822399e+00   1.78974810e+00
   5.67286256e+00   2.16549682e+00   1.95807941e+00   9.01609132e+00
   4.31433495e+00   3.28288153e+00   3.94567111e+00   2.65534373e+00
   5.43694664e+00   9.99063587e+01   7.03571678e+00   4.81329643e+00
   3.00426369e+00   6.75131125e-01   3.67746234e+00   7.01406881e-01
   3.14341836e+00   1.12406233e+01   3.91320991e+00   1.29045666e+00
   5.38893898e+00   3.30223038e+00   1.26626909e+00   4.76306699e+00
   1.35704959e+01   1.49857362e+00   2.23510802e+01   2.14756172e+00
   2.24884946e+00   2.11922285e+00   2.66529296e+00   1.33220692e+00
   2.16163170e+00   1.77178982e+00   9.96146804e+01   8.20181338e+01
   4.60383519e+00   1.10746931e+01   1.23777286e+02   3.26976083e+00
   3.75392738e+00   2.07544860e+00   1.93304430e+00   1.90741473e+00
   2.52455905e+00   2.19968245e+00   4.86642346e+00   2.00086977e+00
   2.41563912e+00   2.32663689e+00   2.03087539e+00   2.51607162e+00
   8.38220530e-01   9.38774801e-01   3.99898849e+00   1.10859048e+01
   2.71888175e+01   9.43840201e+00   1.13562088e+00   1.26412732e+00
   1.78750421e+00   2.54221182e+00   2.61123874e+00   1.56402447e+00
   3.60617217e+00   4.24347493e+00   2.12767519e+00   1.94361639e+00
   1.40398809e+00   1.35802737e+00   2.70431096e+00   2.26094487e+00
   3.88216390e+00   7.12884340e+00   3.57885530e+00   2.09362826e+00
   1.61290098e+00   8.65150749e+00   3.66914966e+00   2.42077534e+00
   1.95650599e+00   5.94615559e+00   6.93794380e-01   5.26315334e+00
   2.54219478e+00   9.61001220e+00   4.71233784e+00   1.50673257e+00
   1.19415823e+01   8.73154966e+00   1.30489101e+00   5.95596788e+00
   3.71663281e+00   1.49203598e+00   9.37017373e-01   4.82606846e+00
   1.56121480e+00   7.51259345e-01   3.23434787e+00   3.00956823e+00
   8.37440804e-01   2.53145500e+00   6.07648326e-01   8.56321419e+00
   1.59692393e+00   1.54725165e+00   2.98690389e+00   4.29763528e+00
   5.01908580e+00   1.81524054e+00   1.06055435e+01   2.03175472e+00
   2.33567650e+00   1.78473895e+00   1.40123890e+01   7.46923230e+01
   2.27076192e+00   6.06970165e+00   8.29992484e+00   1.67404199e+00
   1.84352565e+00   6.04376701e+00   5.73164690e+00   3.68874406e+00
   6.89390718e+00   1.32742497e+01   6.39742398e+00   3.21937940e+00
   6.15423347e+00   3.39299807e+01   1.36881609e+00   9.77559680e-01
   4.27695638e+00   7.95450609e+01   1.03023517e+01   2.58746514e+00
   2.31978133e+00   7.53619267e-01   1.35119362e+00   4.31457248e+00
   2.88141286e+00   1.49521343e+01   8.91441845e-01   2.92074584e+00
   4.55235107e+00   1.99326918e+01   3.79322144e+00   1.00266364e+01
   1.83626023e+00   6.09219120e+01   1.71991117e+00   2.06121657e+00
   4.90301961e+00   3.29913083e+00   1.25907514e+00   6.18358861e+00
   3.68006416e+00   1.24542821e+01   2.22759119e+00   1.02566792e+00
   1.92488392e+00   2.76134696e+01   9.24938739e-01   2.92734657e+00
   3.28386827e+00   1.04868932e+00   1.96848249e+00   3.40508546e+00
   1.90507857e+00   5.67006150e+00   2.72475457e+00   9.49596799e+00
   8.37887501e-01   3.85956919e+00   1.31632147e+00   3.26440713e+00
   2.10767823e+00   1.29430430e+00   2.25833566e+00   3.42166255e+00
   3.77786303e+00   1.85386265e+00   3.10777132e+00   2.34337688e+01
   8.13628655e-01   1.78492314e+00   1.73342126e+00   8.90177062e-01
   1.29910922e+01   1.79883674e+01   1.69240665e+00   4.74700716e+00
   6.75889046e+00   9.58463428e+00   1.52723256e+00   3.15730803e+00
   1.90667851e+00   3.24799062e+00   3.66798570e+00   2.67525895e+00
   1.55437166e+01   3.13942050e+00   2.48848017e+00   1.39623275e+00
   1.17580912e+00   5.20489837e+00   2.23042760e+00   2.49503526e+00
   3.07121268e+01   1.74488133e+00   3.81100404e+00   4.16944147e+00
   3.87196588e+00   2.92257746e+00   2.10061611e+00   1.90798319e+00
   1.43220762e+00   3.43619365e+00   1.48988406e+01   1.20123371e+00
   8.35759017e-01   4.85186980e+00   2.66879432e+00   3.94008156e+00
   1.62619159e+00   1.66629966e+00   8.30193046e+00   2.17513035e+00
   3.08422015e+00   3.05173031e+00   5.16715926e+00   3.21293192e+00
   2.28059680e+00   1.56932350e+00   1.82725768e+00   1.97301963e+00
   1.02276282e+01   1.01957846e+00   1.06828502e+01   1.00808566e+01
   7.91255834e+00   1.01947219e+01   4.80814892e+00   2.17077070e+00
   1.80709703e+00   3.19689929e+00   1.53484010e+00   1.82669039e+00
   2.84114887e+00   3.16190122e+00   8.07517223e+00   6.94925003e+01]
In [78]:
# Mean of means calculations
mm = np.mean(mu)
print("Mean of means:",mm)

# Standard deviation calculation
sd = np.std(mu)
print("\nStandard Deviation of means:",sd)

# Calculating the confidence interval
print("\nCalculating confidence interval using percentile method")
print("95% CI Upper limit",np.percentile(mu,97.5))
print("95% CI Lower limit",np.percentile(mu,2.5))
Mean of means: 9.36128423842

Standard Deviation of means: 51.6119760277

Calculating confidence interval using percentile method
95% CI Upper limit 49.8704090234
95% CI Lower limit 0.751110748119

2.2.5. Repeat it with 3 different N-s.

In [79]:
R = 1000
N= 100
In [80]:
b = np.random.pareto(a=1,size=(R,N)) + 1
plt.figure(figsize=(10,7))
plt.hist(b,log=True)
plt.show()
In [81]:
# Calculating means
mu = np.random.pareto(a=1,size=(R,N)).mean(axis=1)
print("Means:",mu)
Means: [    2.30477062     1.93889435     2.27540887     4.67693512     2.58702041
     6.27588465     4.5460831      7.0846235     21.12794338     5.03492521
     3.25504957     5.20086747     4.70886781     3.55743465     4.39297181
     3.18750835     7.91769729     3.55431667    12.0481028      4.2512006
     2.97614986     2.96907141     5.00728105     5.63670757    25.53680076
     5.25259788     6.35458733     4.92879175    21.71327492     3.56608918
     5.11211545     4.76568244     6.47988264    13.30445873    10.26731723
     2.75509758     6.90679246     6.66695927     2.61047074     5.16094742
     3.46276449     7.05863161     3.51637742     2.13447696     2.14070764
     2.75973912     4.58547128     3.02924644    10.29685597     4.13855685
     3.59664731     4.73075674    18.46235225     3.34048685     4.23110365
     6.73691946     5.76129645     7.87937759     7.30964605     3.74540969
     3.56954345     6.15399219     5.90978618     7.53004486     3.62372453
     6.77489728     7.7266283      5.95554335    27.13513682     3.48264643
    22.99569082     5.54060194     3.28455049     4.83762948     4.34509647
     5.55946131     6.59939926     3.59121209     9.58183051     1.91837659
     7.09753457     2.48893225     7.36500401    13.474932       3.31286261
     2.69392236     3.96969767     2.72783279     8.68850635     4.47208515
     8.45594177     2.67806043     4.02465618     2.78911137     4.63137548
    14.01811284     2.67291348     3.54969065    44.17234436    17.69043961
     5.60937529     4.58088414     7.21159063     3.79608548     4.13120703
    15.3163997      2.47088341     5.63740859     5.80661976     3.83826981
     7.24982827     5.37745026    74.79282685     2.97331995     4.74636075
     3.37091739     5.42326036    15.15045977     3.97689107    17.39885098
     6.60665988     3.1219466     16.53042996     4.22716613     4.55648493
     2.92850716    14.26899036     4.96123323     8.07659171     2.7918315
    19.20854413     4.61311852     3.48706725    10.6876199      3.41113074
     9.54602293     3.81746308     7.11378259     4.79774272     4.26743738
     4.07770441     5.54423982    13.58615968     6.78675958     4.51732
     2.13618359    19.16737046     4.71676727     3.76820047     3.3371125
    15.04311406    39.69246415     4.82660645    25.98329876     8.61621947
     6.65011742    13.31118725     3.42717891     3.89234761     6.56862719
     6.65074924    23.94828002     5.95835582     5.66408828     3.50316871
     4.58928075    16.34635061     5.06787572     6.88586225     4.85329424
     9.8969567      3.57332688     5.85562537     3.75874803     9.73680443
     4.18871758     8.97948533     2.86745434     9.44384818     4.82293138
    18.85807897     6.11667322    16.68131313     7.50688895     3.82742971
     7.27185818     4.10838679     6.07741659     3.69439605     3.13236365
     5.85825238     9.85868413     8.75056374     6.39776184     3.88631878
     3.35573401     9.71125762     3.79853751     2.66370869    45.30774992
     4.0455103      4.17882536    11.98445761     5.40374956     3.61981157
     5.67533331    17.66039051     6.41647899     5.83235972    18.24050296
     4.69511297     3.75630176     3.6528171      2.61226398     8.84237067
     7.17931816     5.60137175     4.40054107     7.73578593     7.16852588
     5.97132679     4.4292451     35.78241035     9.6366822      8.11621656
     5.36855266     5.35066218     3.24333233     7.2622819      4.34144488
     3.8504894      4.81450105     4.19667695     3.61781163     3.66024804
     5.28555392     8.26556275    12.67742714     5.2608985     11.54633051
    95.4470624      8.39411927     6.05126694     2.69911791     6.4682307
    13.5852849      4.15178152     4.69515744     6.97529552     2.60579109
     3.1277495     20.15500013    39.94222582     2.42853376    17.8085343
     2.20936092    15.9589153      2.25968934     3.8621935      6.32449297
     2.81170101    24.89454493     9.395739       5.10906079    55.94361947
     5.42057582     4.26860461     6.15258917     2.27900891    37.1618454
     3.33500229    43.74322584     9.12270989     3.81370392     5.08059921
     2.44798543     3.0482401      3.24466107     7.28686903     5.23629571
     3.105692       9.58485393     6.64798796     4.00527263     3.98027571
     7.8709375     13.08279892     6.6748567      3.42808709     3.87103487
     5.66397884    13.11534708     6.1457679      7.58609711     6.98467469
     2.28668553     3.07875603     5.2406014      8.46513834     5.08998574
    10.21716602     4.65232613    64.13978217     3.0598335      5.08831172
   160.29098202     5.26770573     7.54063785     4.06328073     3.08714242
    12.28632431     7.90095329     6.72335634     6.14763611     4.24076756
    12.41128696    22.98191028     4.4224218      3.52406752     5.70604278
     4.04969477     3.04979844     6.01330352     3.44760423    12.00711869
     5.23711764     3.56182397     2.68316328     5.12354932     5.26620321
     2.88209276     5.83783083     4.90348252     2.79457469     4.75364833
     3.68912234     8.68321348   102.60234341     6.71606182     3.27274459
     2.96354228    10.66546215    10.53791637     4.97658674     4.20413898
    10.99516489     2.55331213    51.37550174     9.69757067     3.33641639
     4.42436012     9.9126732      5.11742953     6.0503109     10.38049057
     4.32387102     6.73355325     8.300137       5.01415248     5.1558709
     2.80526968     2.52185655     7.32971013     2.91579336     6.27430392
     2.30106626     7.41507143     5.44262445     3.73947798     5.07977949
     5.03257463     5.75322912     3.08074191     6.02359372     6.20829142
     2.31822841     3.43912766     5.53854412    26.47210948     2.55307381
     4.2415435      2.91233533    10.8481001      3.72880111     3.16608901
    34.99184352     6.19225636     3.87713238    12.72299798     6.35387915
     5.39809604     7.82492918     2.54838095     4.3768873     14.2407408
    14.44676944     3.86911105     5.14332984     6.71804721    11.11711039
     4.93642534     8.22286578     6.23466162     7.32201856     4.49079023
     8.01875809     3.40260097     3.42801059     5.64161471     8.04579062
    23.81158512     3.92592174     3.69137819     4.25507817     7.92578595
    19.39515051     9.09556837     9.26138964     2.67364998    11.31014153
     5.61210386     8.66921936     7.34967046    14.11213623     5.19821286
     3.02537978     9.46118246     5.71533999     4.2301198      5.01759767
     5.06185502     7.77364354     4.95484636    13.46421999     5.25183694
     3.39439156     3.63702845     5.3898034      2.48465034     3.90584886
     3.58775625     4.55263751     4.59049269    12.01910039     3.6933068
     3.02773206     9.21817453    10.87562864     6.47819069     4.03155908
     7.54396141     4.64125862    19.94296606     2.76657879     3.12203152
     3.18125946     6.82264319  1068.65916986    17.27346927     2.90349278
    21.00360249     4.15691048     6.08653178     7.06231213     3.5042763
     4.87464475     3.80206317     6.26982467     7.7098532      5.14841041
     4.61892087     4.12468594    12.81908469     5.6554771      2.42436195
     5.69052969     6.16656312     4.83665781    10.55574686    21.55672488
     2.63725134     5.55811207    37.84795074     3.6843107     11.7515359
     2.65721376     5.94963274     9.82218694    45.85351473     4.0440459
     2.79944055     4.15999196     3.3551089      2.7402358     13.51582533
     3.32448604     5.68281982     3.27333865     5.29515322     5.01110709
     5.70242272    10.74972327     4.04681046     3.01195061     4.01113141
    14.68643638    15.71505766    28.21231007    15.68405753     9.02852635
     5.50773075     2.40377507    11.31155807     4.01795034     3.68579855
     3.98924767     4.72686763     3.67000566     4.3240368      7.17389183
     6.08875212     4.69142239     3.41866823     3.79822526    11.01623766
     5.53102074     7.42815026     4.06026776     2.9137375      8.80080919
     3.60064689     3.50108964     7.75028217     8.47804481    10.87381922
     4.69206664     8.5491643      4.82835283    10.95173818     4.69276351
    30.5108186      2.99470606     3.94193328     4.48239        8.55378542
     6.87569439     5.11883863     3.61247641     5.5956388      3.83472956
     7.54766625     9.7775015      5.79231887     4.01217069    23.94638652
     5.32348537     9.1437213      2.91534104     2.59725494     5.34106099
     9.86874346    18.82423506     6.46435312     3.45347725     8.71148495
     5.48973072     7.03352323     6.57879848     3.46986126     1.69274909
     7.59482264     5.54535408     2.67962104     3.76630834    31.99067545
    10.67143412     3.17860048     2.83882562     4.0080139      6.19586964
     3.28914326    74.45014124     6.60242609     6.63431084    11.10437947
     2.96868897     4.26493709     4.47331413     4.22624079    51.0689289
    13.42560228     3.28281289     5.87410241     5.88538584     2.77851808
    17.98457949     3.62363081     9.91603324    14.36954815     4.65156355
     4.85516225     3.45200903     3.66563721     8.03620713     4.65816255
     2.50315651    21.77920871     2.54149964     2.8300454     11.89224964
    11.21316672     4.82184376     9.60618885     3.00507585     5.50185931
    12.59321143     3.59481797     4.13696457     5.73037218     3.3460574
     2.22369598     2.45155204     4.20937361    14.06748647     3.95799352
     3.5334394      9.73013315     3.48505779    19.63446262     5.29679914
     2.71730787     6.39669592     6.21734048     3.58029514     3.31524121
     3.81610738     2.67956305     6.92930279     4.08362198     4.90564876
     9.51114623     3.6959228      5.54154823     7.56993577     5.73756263
    10.87948877     7.40042445     2.42657311   218.66521597     3.40559836
     3.19496494    11.40619406     3.69890116     4.20264157    10.20386247
     2.13642021     4.47056657     5.93136984     3.88945937     4.62532061
     6.94498753     3.3413812      5.48400569     8.75280339     3.1232408
     5.31457429     2.64237157     4.02274863     7.53853558     6.48981081
     3.79470787    18.96503507    11.39647531     5.96082554     5.71927829
     7.57136        4.62788552     2.98450822     6.78699736     3.86996522
     9.65796239    13.75976249     3.59799035    13.44385509     6.02126686
     4.88345736     3.51307376     4.08136149    12.77652352     6.43464366
     5.46118374     5.12758416     4.58566147   606.01016855     3.68652762
     6.885458       6.26938236     6.94879324     4.64080928    16.82140249
     6.58892269     3.42738979    10.77930608     8.13133004     3.44029757
     4.31675533     2.7721838     29.53597043     4.41006726    11.54715877
     5.45305187     9.75617337     5.57822393     5.36620771     2.7313244
     3.57661177     3.16615983     3.21788668   230.84382464     4.07086762
     4.83613647     4.75647299     4.06114167    29.12707756     7.22873888
     2.96646055    13.89988572     8.38297638     2.36834845     4.32290768
     5.55759354     4.29308103     5.19486112     7.11235342     5.98453524
     5.95856041     4.13088526     7.90818023     5.9404983      3.46909209
     4.79828648     5.87975826     4.36383723     6.05631031     4.38066467
     9.18207609     2.48596827     2.73278243    23.30625463     6.30144955
     8.33328399     4.48070597   301.49284281     4.71811208    27.41899556
     3.95006034     6.08564193    25.68308624     9.27222281    44.2721626
     2.92165537     5.96080836     4.60677052     6.08531286     3.53175602
     6.47973317     5.49026653     7.61868626    15.63349813     8.71857414
     7.87880684    30.32106832     5.8658671      4.58982487     5.96554305
     5.83775788     8.57240581     3.63562697    38.73674856     5.59375887
     3.13603941    17.56048319     7.73102636     2.68838606     8.81651155
    14.95105513    26.00571034     2.88270788     3.2916062      4.47084417
     3.56607593     4.36370808     2.46014011     5.42854764     6.26402481
     5.58024489     4.52875384     9.97384581     2.69245143     3.16410305
     5.56391734     2.47801094    20.32605183    17.6807214     40.34109445
     5.09912673    16.08923077    39.5119474      6.79370941     8.99344766
     4.39032636     8.91845532     6.71764176     4.039877       7.73032623
     3.47978953     8.16508691     6.79169156    25.02637857    15.35221202
     8.50065617     2.82830314     2.85030025     7.32737934     5.56087136
     7.56275986     4.38639739     3.02892817     4.47529334    47.58520189
     3.4270822     18.4940009      6.54158567    18.08876656     3.19497229
     7.40592021    25.44165293     2.6137032     17.31899092     4.21209413
    10.80486703   272.58282514     4.81067919     3.14652205    10.71173271
     5.20001266     6.9964165     10.89739684     4.76093063     3.99527354
     3.22302519     4.56939904     6.48838372     4.28604641     2.32642614
     1.92581644     9.01078465     6.63132022    21.27978155     3.26291521
    14.4141087      2.97252023     4.76082365    45.29370488     4.0289862
     7.51821456    18.47846338     3.8661139     22.71163936    10.58813519
    22.7204838      4.06686516     5.33588719    10.84290005     5.66689061
    68.71771374    10.17796819     3.66162284     9.45678349     1.92553015
     7.54744777     4.51706434     2.83126657     3.8483684      4.5501154
     2.3792245      3.36155009     9.09820173    14.63693773     5.58963404
     2.94525924     9.08490794     4.01809708     5.27550472     5.89818221
     2.5892982      3.68340501    35.30692102     6.09403284     5.69123286
     2.77956371     3.61223718     2.995994      23.66785132     9.77077613
    11.64631855    11.14682657     4.39434674    10.00754958     7.33176232
     8.08307847     4.56698893     5.82977178    10.1030807      6.47839753
     2.95998414     3.13940967     6.76170234    10.25377968     8.30017651
    16.55162511     5.46639926     6.00395346     2.16011914     6.8852539
     5.62281262     4.19008827     6.48540879     4.0850207      3.62474941
     4.72040411   321.84501178     4.89285104    56.03998203    16.27644612
     6.12817157     3.46368976    99.35881816     3.95327577     3.72982923
     8.55076297     5.15274377     9.63943691     7.78121121     2.40488267
     5.76983987     4.13905219    20.44023253     5.0333332      9.21680253
     3.83412109     3.83914172     4.90183793     3.36868401    19.16391568
     6.37937607     5.58434226     3.67703334     5.70614593     6.84526379
    12.3300209      2.99464383     8.9501112      7.82105596     3.74897307
     2.81394916     3.82164744     5.26976279     6.3366103      8.02090953
     8.53958081    10.78716739     9.24582485     6.64425848     5.86859855
     6.46614408    20.47814487     6.02872442     8.27556656     4.38723755
     6.05336353    28.6227008     39.96974245     5.51412169    11.61333611
     4.17416018     5.1456197     35.4176473      9.30546512     9.60113977
    17.52199469     3.58824313     3.96332596     5.21990844     4.61869301
     3.6839839      6.90567022     5.93441352     5.2497864      6.13567219]
In [82]:
# Mean of means calculations
mm = np.mean(mu)
print("Mean of means:",mm)

# Standard deviation calculation
sd = np.std(mu)
print("\nStandard Deviation of means:",sd)

# Calculating the confidence interval
# Calculating confidence interval using percentile method
print("\nCalculating confidence interval using percentile method")
print("95% CI Upper limit",np.percentile(mu,97.5))
print("95% CI Lower limit",np.percentile(mu,2.5))
Mean of means: 11.3797984553

Standard Deviation of means: 44.0193848613

Calculating confidence interval using percentile method
95% CI Upper limit 43.7539538067
95% CI Lower limit 2.42651783017
In [83]:
R = 1000
N= 500
In [84]:
b = np.random.pareto(a=1,size=(R,N)) + 1
plt.figure(figsize=(20,7))
plt.hist(b,log=True)
plt.show()
In [85]:
# Calculating means
mu = np.random.pareto(a=1,size=(R,N)).mean(axis=1)
print("Means:",mu)
Means: [  269.58898942     9.15989342     5.89476372    19.5010269     41.42513737
     6.50481166    11.07836322     6.49393482     9.21801979    12.04076439
     5.75213808     5.83988141     6.84798352     7.0625811     14.06112566
     6.90904584    46.7752508      5.93820374    10.5023852      8.70517785
     5.72457479    10.33066502     5.19836759    13.34484291     3.77086647
     4.67230523     3.9496583      4.32890578     5.10343262     6.12302149
     5.00175492     5.32831056     6.68632375     6.32596133     5.26959485
     7.32473801     5.52062848     5.05101271    13.74301709     8.26219594
     6.21028121    11.33126431     7.37358608     4.74193069    11.14208069
     4.4507803      6.26244971     6.97079272    12.45881426    11.65827688
     7.21559567     6.79929719    19.17621483     5.9924374      5.86888127
     4.7049783     13.82816468     8.94896129    94.07052321     9.80728589
     6.364938       5.28338957     5.56684157     4.50038423    31.65468169
     4.65160178     7.49915326     5.17035692     7.10203498     7.47912593
   477.81760632     9.26315297     9.82260433    10.0776329      6.29175663
    13.0387543      9.45690491    20.43340793     4.29667283    16.85644921
    42.73748908     7.17757275     4.75942646     8.21725276     5.94735732
     7.1690476      8.59854161    12.99484711    11.26745338     6.08555168
     4.40113242    10.66391848     5.03822552     4.05961245     9.97742068
     3.97712754     8.45737884     6.75953483     8.02041352     9.39910116
     6.04859256     4.43457334    19.44403687     4.11033969     7.30248303
     5.49781381    30.0522794      6.00143029     7.79943717     9.84617321
    10.97573392     6.18581849     5.24078883     5.15431721     8.86751153
    35.40592512    13.54528639     5.90972253     5.00763767     7.78231866
     5.94829736     5.08262381     6.70240311    12.05245615     9.64090251
     5.67276465     6.95658184     5.70516004    23.62578142     6.25162548
    24.77438587     8.57045464     5.72960839     6.54962586     6.23596864
     5.61288296    56.50094672     6.38872515     5.8434561      6.96986011
    18.07239727     4.92636861     6.81016936     8.56603994    35.50238877
     5.41957722     6.1822805      8.70813513     7.41704009     4.36617397
     7.26112142    12.1646305      5.82081185     5.26656658     5.9665828
     9.11353664     9.87097974     5.14587727     3.84665502     6.10592968
   367.44791665     9.29441848     5.31289047    14.03253746     4.99100646
     4.10837545     4.95720074     5.61423294    10.26500751     7.50786906
     6.92883974     7.59730349    18.7734753      6.10500489     8.27536685
    10.30968673     5.69906178     7.11896018     9.69213741     4.89944731
    13.36919141    11.29413346    13.01953363    12.24632465     5.80052626
     5.8690934      5.75398288     8.10529623    11.65587703    17.75951573
     6.65859966    75.09592838    22.78057253     5.34999135     4.60345737
     6.41881282     4.21410781     4.72395536     4.70440681    14.19948779
     6.65475        7.07384942     5.83100175     5.33719343     7.84927865
     4.91192393     5.27757402    12.20043207     4.57520861    46.92348769
     8.26810833     7.21326448     9.73957821     5.7817333      7.37005207
     6.2504866     42.95435891    11.48998136     6.35759999     4.76779757
     6.27899034     6.39916196     6.66465969     5.06265049     7.32262094
     4.92866853     6.01317037     4.42442223    21.11863166     4.44722482
     4.23205723     3.75201858     9.70357623    60.56488911     5.23129699
     7.94127607     4.99551823     4.84940118     6.15751063     7.70246391
     9.27195868    17.89340904     8.06052952    16.03672961     5.69826423
     3.91851491     4.8972774      7.02537091     5.70610148     6.51267653
     8.3040344     18.22102747    16.33320323    11.89740424     4.50852547
     4.66519964     5.28261008     9.49971801    11.89627128     5.64140132
    11.09800196     4.86852768  2335.21624866     6.61408169     5.39850205
    10.65851513     5.53813253     9.21721887     6.88438068    17.04038384
    12.91447924     6.23863526     4.49068357    13.97040385     7.30674165
     8.06103269    10.51734392    62.47308048   131.65671447     8.20189537
     4.4770043     13.37366423     7.74035841     5.10568749     8.82399159
     5.86522921     6.47219336     9.17904799     4.52254862     7.92105583
    16.56059095    10.20694892     5.45863495     6.02182691    10.93839413
     5.86327691     8.02153909     5.59239603     5.85943363     7.20047268
     7.10844905     5.92471464    45.61997733     5.85312708    16.94484624
     5.54899802     5.53181394    17.6595806      5.6955923      7.47635206
     5.2711342      4.24637671     6.13086995    30.45066579     3.29327664
     3.67349325     8.03204378    11.35051772    26.05379896     4.84030412
    24.76553591    11.06097202     5.01581061     7.71711256     3.06791378
    25.94990366    28.09229715    14.12399486     6.37449299    17.1536218
     9.84306597    14.27193308     6.78590274    28.38755212     8.09394762
    24.67569557     5.42115935   275.68423229     6.24365073    11.58470308
    35.1922662      4.49249821     4.16810098     5.56388542     6.47079373
    32.9251562     11.70246501     8.17571736    16.92415177     5.61481433
    10.73522065     4.85737279     6.4223559      8.37042752     5.40147027
     4.71340048     7.84893786    12.04931838     7.70817143     5.20795179
     7.70091694     5.08381423     6.86035632    37.4147777      4.90891617
    13.06371712     8.42994552     5.78980909     5.13385154     7.29178134
     7.22684383     8.56381518    13.11504772     9.57160836     4.07100408
     5.72325114     6.6298264      6.11186881     6.77477513     5.84105805
     6.2648869     11.10092557     5.38865673     8.64901485     8.52199518
    12.63235185    10.99866096     5.2995408      4.11217652     4.17367511
     6.6420151      6.87512187     5.44191197     6.76653443     3.36170127
     5.05110947     4.76145942     6.3400848      6.61790876     5.74124891
     7.6203998      7.62715712     4.92462231   105.51278362     6.6463066
    20.06824446     4.33611165     5.27054562    14.11749586    11.93094637
    25.64380479     4.59752284     9.35447635     6.87092895    19.60702163
     4.36331033     5.45597987     4.73689093     5.81243193     6.24732472
     9.81836207     6.14295653    15.25095183     6.51121966     6.12195598
     5.93791164     7.53283779     7.02898103    14.26291454     8.15292081
     4.00129467   124.97180978    97.95923247     9.08218288     6.28639568
     7.01030476    18.49061763     9.23732366     6.07969513     5.96612667
     6.34173753    12.2193778     17.59681251     8.56106055     4.1523311
    18.0544719      5.73460596    16.5103759      7.13820263    14.27045618
     4.6297781      7.27559692     7.2909116    128.14150251     4.05928402
     7.14311412     8.9149217      7.23264846     7.25947901     6.61175486
     7.10523183     6.71788854     4.94413589    13.47046857     5.37268221
     4.76135885     7.87597846    38.1772896     11.08241013     5.57305248
    16.88526203     7.59333981    38.42294813     9.78011549     6.71947226
    16.26641171     7.22659301     5.38723583     6.95641183     5.4394867
     7.33787894     5.61251265    16.63717355     6.98489893    13.14591242
     7.11319642    11.36253036     7.37151004     5.22905183     9.05671052
     4.54325902     4.79588366    11.62930595     6.04114702    43.39637038
    36.50316764     7.28978624    13.75539957     9.40691539    14.89146179
    10.48782538     8.76467928     4.84248203    14.61909857     6.49869707
     6.22337078     9.52255804     6.19175004     8.0344999      4.05466174
     5.86962328     6.59593948     9.19189054    40.2146467      6.62007375
     4.92959357     7.97507991    14.92640652     5.14277364     5.54596035
    36.76684104     5.3537428      4.8012409      4.6250391     15.51514654
    10.11672941     8.37905592     4.64366656    11.50780809     5.7541249
     5.5048028     10.18827951     6.11774891     7.14266085     8.55665202
    41.42603485    17.61895278     8.01524478     3.72075114    14.21944416
    10.71361586     6.6422219      9.88143121     4.22631168     4.14390469
     5.51527296     3.63554502     5.45615569     6.53593277     4.71299877
     8.57631968    10.49627082     6.43093016     4.62758855    15.02103514
    94.47726437     4.61153455   183.95544775     8.82416459     7.4849667
     4.12933115    10.58776472    14.74979255     6.10739544     4.08896116
     9.96857088     5.21202786    10.16162722    25.94012267     4.24355255
     3.70179941     4.26890525     4.20355281     6.36575899     6.63999088
     9.03726327    11.52208588    33.90727701     6.78830436     3.88789794
     7.58049275     7.23727403     3.9353125     11.14591004     7.26452066
    68.20328129     4.69401265     6.42129486    10.97576229    10.12226176
    11.62845802     6.32197151     9.04632956     6.91613587     4.12043409
     6.70457567    24.08388869     5.50072256     8.20580628     9.20397785
     6.85434877     4.95341581   563.19703781     7.18988671    13.42410346
     5.86556256     4.73600423    20.2667955      5.78741949     7.99748113
     8.3255435      7.01961424     5.09065346     9.41911835    10.87461124
     4.10923005     4.03734798     5.77158278    13.58362555     6.83859688
     5.71180312     6.21239211    20.38102057     5.52612461     6.57415649
     5.03373181     6.84218268     8.12049311     7.07555155     8.72647302
    15.99152544     8.76453709     7.90777967     8.29039308     4.6858375
     8.16367915    19.6121543     12.00734005     8.83702943     5.34205051
     6.42753701     8.16327201     7.0038848     10.16517336    19.15948668
     6.14010234     5.16702547     6.17268238     8.41593804     4.79154391
     3.90683296     4.27258964    12.5088594      7.8096059     12.77411316
    30.06554582     4.95469774    46.28610363     8.7842152      6.86378931
    20.52653281     9.02344116     4.2042468      4.48024399   178.55891467
     6.94119358     5.04836388    51.16178284    10.54189014     4.71718281
     9.48672438     3.16683427     6.3579188      9.06897817     5.47864269
     4.99514657     6.81043431     5.29462824     6.32066379    30.9072142
     4.28272901    69.19168437     7.99567212     5.66823594     6.55086931
     7.97838635     6.77069776    31.10649379    11.63662712     5.39329115
     8.25094315     6.81258158     6.0739915     13.04745259     5.64815823
     4.08684119    18.82881003     5.92504955     7.92281663     4.72652215
     6.58713134     5.89889574    13.75040708     9.29593928     8.21938972
     5.19556757    19.10830589    19.83230576     6.29111991     4.5013779
     9.35010839     6.88702034     6.97460663     9.55723193    12.07630874
     7.73332195    10.33942194     4.66674136     6.07744133     8.76093043
     5.47814341     9.78572845    20.92484576     3.77405049     8.04236419
     5.86171958    19.9508135      6.05062244    10.40106144     5.09774923
     4.26242616    28.59012746     4.49641311     9.89614575     8.79224449
     7.46703949     8.80260144     7.51583605     6.20629851     4.53148343
     8.6644768      6.09838978     5.18082756     4.07087104     6.52625417
     4.66716965     7.293893      18.3301304      5.8718602     19.59356671
     3.9919764      4.96603665    10.01027492     4.17404381     6.91859616
    20.28647738     5.95336506    39.34561048     3.64472268    10.08508644
     5.29534604   234.12899283    10.46844852     5.86552976     5.81406138
     6.10513893     8.11402447     5.15699084     6.22043485     4.53323974
     5.34554564    13.05737558     7.60407651     5.6521739     84.21429024
    11.58594041    15.68490161     5.98376393     6.45456843    16.67432196
     7.50745232     3.24350951    13.56324582     8.02139845     5.18209621
     3.74405912     5.82043903     5.00635924    17.87315881   193.78184855
     5.39968574     6.84838981    15.51691801     5.35800485    32.19226209
     5.74755272     5.6608278      5.80076405     6.14155005     9.04967077
    32.65062668     3.45882885     5.29115942     6.80134171     6.94174146
     7.11093541     4.56452692     9.55505915    11.0406491     21.09209879
     6.22254624     9.31596593    10.29401836    11.483354       8.17427502
     4.45986001     4.38029965    25.33595466     6.16154229     3.53820345
     9.52311108     4.98611473     9.13413609     3.22458495    16.29794102
     6.81389632     5.98700192     4.86856779     5.75675665     5.93192055
     9.01480334    26.82044235     5.31150782     6.03811261     9.34515769
    25.63837847     6.35563875     7.87190721     7.10494503     6.29316957
     4.76421136    18.557233       7.52966053     4.24635762     9.88166325
     9.81897547     8.00484484    11.88823555     8.30442103     6.17815812
     5.55684835     5.43246266     8.5850352      7.77522704     6.97673883
     6.07672605     9.75931905     4.46055164     4.17784438    13.78619615
    13.00606331     7.64379029    10.13239588     4.81026379    21.8380589
     8.939085       5.25227118     3.97009817    20.70784263     5.36504086
     6.59220289     4.10014993     6.39987594     5.03556859    44.36623113
     5.6734599      9.98400465   791.69819833    11.98595575     5.10456829
     6.50990545     4.78069774    29.32479224    11.3503268      6.90858338
     5.91913865     4.6508854      7.15072148     8.76803474     6.20537637
     9.99894785     5.73288495     7.98200519     6.00179065    11.57342887
     7.56190149     5.13116449     3.73523096     5.43885764     4.27353673
    16.90088216     6.44794472     8.60247319     5.29141517     8.30304267
     5.60860048     5.88708819     7.44849645     6.28076199     5.40162116
    10.62501        9.84300888    27.89318161     3.64569905     5.84119252
     7.02999149     6.78479071     6.67479792     8.27800732     4.78955757
    11.36829375    18.89622535    13.36005094     7.38029441     4.45405016
     5.97017055     5.67484773    35.42054295     5.86745005     7.11963415
     5.80512847     5.31918461     6.58310496    11.15853499     4.85069002
     8.54416068    16.58954705     4.86500083     6.25472415     9.50674713
    34.67007788     6.73621194     4.82270292     9.15177089     4.6697987
     6.36330448     7.68324586     4.47078542   252.86248683    33.7642996
    19.56444586     5.16725222    13.49324879     8.15726521    38.99513751
     5.25067567     6.51884721     7.087761       6.16489558    42.24126173
    44.46243511    15.67154106    12.2942973      4.18080983    55.38032005
    22.27146694     7.56969464     4.94008306     9.66228214     7.6353201
     5.61314985     6.53978923    21.2976741      6.64497483     6.67938915
    52.0454462      4.55455099     5.37907539     4.66889416     5.91681088
     4.35502296    55.76436633     6.83065974    15.09974604     9.15415795
     4.20818215     2.87456925     5.04683624     8.76125477    11.21985891
     3.70787273    33.65211741     4.32351927     5.93851887     9.61619043
     7.06521046     5.45324166    10.09585746    20.6325984      6.62170434]
In [86]:
# Mean of means calculations
mm = np.mean(mu)
print("Mean of means:",mm)

# Standard deviation calculation
sd = np.std(mu)
print("\nStandard Deviation of means:",sd)

# Calculating the confidence interval
# Calculating confidence interval using percentile method
print("\nCalculating confidence interval using percentile method")
print("95% CI Upper limit",np.percentile(mu,97.5))
print("95% CI Lower limit",np.percentile(mu,2.5))
Mean of means: 16.7097797473

Standard Deviation of means: 84.3839777325

Calculating confidence interval using percentile method
95% CI Upper limit 56.60254528
95% CI Lower limit 3.93489256039

2.2.6. Finally compare your results for normal and Pareto distributions.

Based on the plots and results above, we have following observations:

  • In normal distribution, the value N affects the result we obtain for the mean of means, standard deviation and confidence interval values. The higher the value of N the smaller the mean and Standard deviation values and the narrower the Confidence interval.
  • In pareto distribution, the results obtained for different values of N, didn't quite follow a defined pattern. Sometimes, the mean increase when we moved from N= 10 to 500 and sometimes it reduced for N=500. Similar pattern was seen for the standard deviation values. Sometimes, the confidence interval reduced moving from 10 to 100 however for larger N value the confidence interval increased while in some cases for the same values of R and N, it increased with increase in the value of N.